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THE NEW INFINITE AND 
THE OLD THEOLOGY 



By 
CASSIUS J. KEYSER, Ph.D., LL.D. 

Adrain Professor of Mathematics in 
Columbia University 




New Haven: Yale University Press 

London : Humphrey Milford 

Oxford University Press 

MDCCCCXV 






Copyright, 1915 
By Yale University Press 



First printed July, 1915, 1000 copies 



/^. 



iC 



JUL 24 1915 

)GI.A406978 



i 



PREFACE 

Some years ago, in the course of a lecture 
dealing with Mathematics regarded as a dis- 
tinctive type of thought and with its rela- 
tions to other varieties of philosophic and 
scientific activity, I ventured to say: "I do 
not believe that the declined estate of The- 
ology is destined to be permanent. The 
present is but an interregnum in her reign, 
and her fallen days will have an end. She 
has been deposed mainly because she has not 
seen fit to avail herself promptly and fully 
of the dispensations of advancing knowledge. 
The aims, however, of the ancient mistress 
are as high as ever, and when she shall have 
made good her present lack of modern edu- 
cation and learned to extend a generous and 
eager hospitality to modern light, she will 
reascend and will occupy with dignity, as of 
yore, an exalted place in the ascending 
scale of human interests and the esteem of 
enlightened men. And Mathematics, by 
the inmost character of her being, is espe- 
cially qualified, I believe, to assist in the 
restoration." 



iv PREFACE 

The following pages have been written 
under the stress of that conviction, which 
the intervening years have but deepened and 
confirmed. Rational theology is a legitimate 
and venerable member of the great family of 
spiritual enterprises of man: natural science, 
philosophy, jurisprudence, religion, art, 
mathematics, theology. These are all of 
them children of one great passion: the 
imperious craving of the human spirit for 
an inner ideal adjustment of life to the 
tragic limitations of life in a flowing world. 
The distinctive problems of rational theology 
are regarded as in a special sense originating 
in what may be called the supernalizing 
tendence or power of the human mind. This 
propensity or power, so strange and so famil- 
iar in every category of the understanding, 
ever and everywhere manifesting the pres- 
ence of a kind of divine energy in the 
world, is a ^natural' agency, being at once 
a human faculty and a cosmic force, deeper 
than will; and so rational theology is con- 
ceived to be a species of ^natural' science — 
that branch of it which has for its special 
task to study and to appraise the phenomena 
of Idealization. 



PREFACE V 

The aim has been to set the matter in the 
increasing light of certain ideas and methods 
of modern mathematics. But the reader 
need not be deterred by any fear of tech- 
nique. All that is required is a fair share 
of mathematical spirit, which is a pretty 
common possession, being simply the spirit 
of right thinking, or logical righteousness. 

I have to thank the Editor of the Hibbert 
Journal for permission to employ here, in 
some instances with only slight change, a 
few considerations adduced by me in an 
article published in that journal several 
years ago under the title, "The Message of 
Modern Mathematics to Theology. 

Columbia University 
April 15, 1915 



99 



THE NEW INFINITE AND THE 
OLD THEOLOGY 



THE NEW INFINITE AND THE 
OLD THEOLOGY 

It is the aim of this essay to show that 
the modern concept of infinity together with 
certain kindred ideas that have come into 
mathematics in the course of the last hun- 
dred years have qualified this science to shed 
new light upon some of the harder problems 
of rational theology. No demand will be 
made upon the reader's knowledge of mathe- 
matical technique; all that is required is a 
fair measure of mathematical spirit, which 
is simply the spirit of logical rectitude. 

The reader is entitled to know at the out- 
set that the following words are not those 
of a professional theologian; they have no 
official authority, nor any merit beyond what 
may prove to be their reasonableness ; they 
are offered as the words of a layman who, 
in his earlier and more expectant years, 
listened attentively to some hundreds of 
sermons, who has diligently read some the- 
ological works, and has reflected a good deal, 
not without some temperamental interest in 
the themes, upon the great questions that 



2 THE NEW INFINITE AND 

attend a poignant sense of the world's mys- 
tery and wait upon the leisure hour and the 
pensive mood. 

The problems are many and difficult and 
old. No one who has seriously reflected upon 
them or is familiar with their history will 
expect to find in these pages a universal 
resolvent for theological difficulties. Prob- 
lems that triumphed over the keen and 
sanguine dialectic of the ancient world, 
problems that baffled the infinitely subtle 
genius of the middle age, problems that the 
profoundest meditations of modern philoso- 
phy have not been able to solve, present grave 
difficulties. Many of them not even the 
adventurous spirit of modern mathematics 
may confidently assail. My task is limited 
to showing that some of them may be partly 
or wholly overcome by mathematical means. 
That all the rest may be subdued in future 
by similar means I do not maintain. Who 
knows? It may be that some of the difficul- 
ties are insuperable and so are destined to 
be everlasting. In that reflection there is 
nothing to bewail. One need not have 
"passed on life's highway the stone that 
marks the highest point" before learning to 



THE OLD THEOLOGY 3 

be content with less than the full measure of 
intellectual conquest dreamed of in youth. 
To be happy it is not necessary to conquer 
the invincible; it is sufficient to advance a 
little where progress is possible. Indeed it 
would be a matter for sorrowing if in the 
course of time all problems were solved and 
questions ceased to be, for a world without 
wonder were a dreary place. But of that 
there is no danger. Wonder increases with 
knowledge and knowledge with time. "It is 
no longer true," said Henri Poincare, "that 
there are solved problems and others that 
are not solved; there are only problems 
more or less solved." As with natural 
science and mathematics, so too with phi- 
losophy and theology : not complete solutions 
of their problems, not final answers to 
the deepest questionings of the spirit, but 
ever increasing illumination of them, the 
acquisition of fresh viewpoints and new 
perspectives — the advancement, in a word, 
and multiplication of insight and vision — , 
these, I take it, are the reasonable expecta- 
tions, the precious fruits, the ample rewards 
of serious speculation. 

This, then, and not any magical formula 



4 THE NEW INFINITE AND 

for the solution of riddles, is the kind of ser- 
vice that rational theology may expect from 
mathematics. I am aware that, owing to 
the popular misconception of mathematics, 
the claim is not an easy one to vindicate. To 
the many who are accustomed to regarding 
mathematics as merely a useful drudge, the 
claim will naturally seem to be groundless 
or visionary. But their conception of the 
science is far from adequate or just. Mathe- 
matics is indeed a humble servant — a drudge, 
if you please — an unsurpassed drudge — in 
the sense that nothing else does a larger 
share of humble and homely work. To 
imagine, however, that her place in the 
hierarchy of knowledges is thereby defined 
is hardly the beginning of wisdom in the 
matter. It is necessary to look much higher. 
Her rank in the ascending scale is not that 
of a useful drudge, immeasurable as is her 
service in that capacity; it is not merely 
the rank of a metric and computatory art, 
invaluable as the latter is as well in science 
as in the affairs of the workaday world; it 
is not even that of servant to other sciences 
in their fields of experimental and observa- 
tional research, indispensable as mathematics 



THE OLD THEOLOGY 5 

is in that regard; over and above these 
things, she is charged with a sacred guard- 
ianship — in her keeping are certain ideals, 
the ideal forms of science and the standards 
of perfect thinking; she is concerned, not 
with the vagaries, but with the verities, of 
thought, with select matters independent of 
opinion, passion, accident, and will ; it is thus 
peculiarly hers to release human faculties 
from the dominion of sense by winning alle- 
giance to things that abide; her meditations 
transcend the accidents of time and place; 
it is their idiosyncrasy to have for subject 
proper, not the fickle and transitory ele- 
ments in the stream of a flowing world, but 
those aspects of being that present them- 
selves under the forms of the infinite and 
eternal. 

It will be a useful preliminary to reflect 
a little upon the relations of rational the- 
ology to religion on the one hand and to 
science on the other, with a view to ascertain- 
ing what the province of theology may be 
rightly said to be. What, then, are those 
relations? It is evident that the answer 
must materially depend upon the relations 
that science and religion themselves bear to 



6 THE NEW INFINITE AND 

one another. This subject I have discussed 
elsewhere. In a recent lecture* dealing with 
science and religion I have undertaken to 
examine the relations between these two great 
interests of mankind from what may perhaps 
be regarded as in some respects a new point 
of view. I cannot here repeat the considera- 
tions adduced in support of the doctrine 
there sketched. It will be helpful, however, 
and possibly sufficient, to set down briefly 
some of its cardinal propositions. Those 
most intimately related to our present enter- 
prise are these: 

A. — In respect of method, structure, and 
content, science is conceptual and logical. 
Any branch of science, at any given stage 
of its development, consists of a certain 
group of ideas, or concepts, together with 
the relations that bind them into a logically 
organic whole. The potential domain of 
science and the domain of the rational — 
whatever is open, that is, to conquest by the 
means of concept and logic — are one and the 
same. All else — whatever is below or above 
that domain — is subrational or superrational. 

* Science and Religion : the Rational and the 
Superrational. The Yale University Press. 



THE OLD THEOLOGY 7 

B. — Religion, on the other hand, is not 
essentially a body of ideas nor a body of 
ideas together with their interrelations. 
Religion is essentially and ultimately a com- 
plex of emotions, of emotions as felt in their 
integrity. It is thus a kind of life not known 
nor knowable conceptually, logically, ration- 
ally, scientifically; it is known or knowable 
only "emotionally" and is even thus know- 
able, like love, for example, to none but such 
as feel or have felt the constituent emotions. 

C. — Religion does not belong to the 
rational domain. There is indeed possible 
a science of emotions but these can not, as 
emotions, be constituents or elements of it. 
For it they do not exist as feelings. It can 
know only their outward manifestations and 
can know these only as science may know 
other objects of the external world. What 
is called the scientific study of religion does 
not — as scientific it can not — deal with reli- 
gion as "emotionally known"; it can not 
know religion as a felt life, as a life conscious 
of itself; the most, the best, the last it can 
do is to know, as objects, as externalities, the 
exterior manifestations of what is essentiallyj 
being emotional, an inner life. Concepts can 



8 THE NEW INFINITE AND 

not feel, logic can not fear nor love, it can 
not revere, wonder, worship, nor adore. For 
scientific method religion is not a life, it is 
an hypothesis. 

^- — The doctrine that it is a character- 
istic mark of religion "essentially to deal 
with the uncharted region of human expe- 
rience" is untenable. Ignorance is not the 
presence of religion — else every body would 
be profoundly religious—, it is the absence 
of knowledge. Religion and the spirit of 
science are not incompatible; being capable 
of dwelling together harmoniously in a single 
personality, they are compatible practically; 
and they are compatible theoretically: under 
the influence of advancing science, forms of 
religion age and pass but new forms succeed 
them, and the religious emotions change but 
they do not die; in this respect it is with 
religion as with knowledge — there is trans- 
formation and supersession of form, there is 
advancement, enlargement, and elevation, 
but no breach of continuity, no essential 
extinction, no death. 

E- — The rational implies and in a measure 
reveals the superrational. The rational 
world — the potential domain of science, the 



THE OLD THEOLOGY 9 

field of concept and logic — is not the whole 
sphere of our psychic life. It is but a mid- 
region, the median zone; under it lies a sub- 
rational zone — the zone of sense, which we 
share jointly with the beasts; above it, a 
world superrational, which millions have 
fancied angels share with us. Though it is 
above and beyond the dominion of concept 
and logic, the existence of that world is yet 
betrayed and its nature in part displayed, 
by rational means : by a process known in 
mathematics as the method of limits but 
elsewhere known as the process of idealiza- 
tion. Operating amid the activities of con- 
cept and logic and upon their subject- 
matter, the great process occurs in every 
division of the rational understanding; its 
function is, in every category where the laws 
of reason reign, to point aloft to an appro- 
priate limit beyond their range, to some ideal 
form above the laws: in the category of 
classes, to an ideal universe as the manifold 
of all; in the realm of propositions or that 
of relations, to the sum or the product of 
all propositions or all relations ; in that of 
time, to eternity; in knowledge, to omnis- 
cience ; in ubiety, to omnipresence ; in power, 



10 THE NEW INFINITE AND 

to omnipotence; in order and law, to 
necessity or fate; in indetermination, to 
absolute freedom or self-determination; in 
wisdom or love, to the "beauty absolute" of 
Plato's dream; and so on and on throughout 
the circuit and scope of rational thought. 
And so it is that the realm of superrational 
reality — the ultimate source of the religious 
emotions — thus indicated by the supernal- 
izing process of idealization operating in the 
fields of reason, presents itself as an over- 
world of ideals. 

In view of these considerations respecting 
the relations of Science and Religion, what 
shall we say is the place of Theology ? What 
are the essential relations of Theology to 
Science and to Religion? What and where 
is the province of the venerable "Queen"? 
It is evident, I think, that theology has a 
province. Certainly there is in the heart of 
mankind a perennial craving for a kind of 
wisdom that the ages have taught us to 
regard as the peculiar object of theological 
aspiration; and there is a corresponding 
realm of truth: a field of enquiry that the- 
ology may rightly claim as her own. We are 
in a position, I believe, to see pretty clearly 



THE OLD THEOLOGY 11 

what her province is and what it is not. I 
speak, of course, of rational theology. No 
one need be told nowadays that theology is 
not religion. Religion, we have seen, is essen- 
tially and ultimately a certain complex of 
emotions — of emotions, not as analyzed into 
their elements, but as felt in their native 
integrity. Religion, so taken, is not only 
more immediate and more fundamental than 
theology but differs from it in kind : theology 
is not emotion, it is doctrine. No doubt 
religion is, in a sense, pregnant with the- 
ology, containing it, so to speak, in a "state 
of solution", in potentia; theology thus is, 
in a sense, religion's offspring and is natur- 
ally pervaded and tinged by religious refer- 
ence and feeling. But to confound or to 
identify the two things would be like confus- 
ing a doctrine of sesthetic with the sentiment 
of beauty, or an ethical theory with the 
sense of right and wrong, or mathematical 
science with a feeling for logical implication 
and intellectual harmony, or science in 
general with the feeling of wonder, the 
delight of understanding, the lure of truth, 
the joy of knowledge and light. All doc- 
trine, all theory, all science results from the 



12 THE NEW INFINITE AND 

reaction of intellect to feeling. "GefuhV^ 
may not be "alles^^ but it is back of all and 
under all. The emotional source, however, 
or background of a doctrine is not itself 
doctrine. A dogmatist may feel, but dogmas 
are not emotions, they are propositions. 
Rational theology, in order to be rational, 
must be an affair of intellect, it must be an 
affair of ideas and their relations, of con- 
cepts and logic ; it must be scientific — scien- 
tific in subject-matter, in method, and in 
structure; and so must deliver its message 
in the form, not of poetry or song or ejacu- 
lation, but of reasoned propositions concate- 
nated into an intelligible and coherent body 
of doctrine addressed primarily to the 
luiderstanding. 

If theology is to be thus regarded as a 
science, what shall we say is its subject- 
matter.^ What is theology the science oi? 
The answer is hardly to be found in the 
etymological meaning of the term. Names 
are stabler than their meanings. Time is 
ever pouring new wine into old bottles but 
the bottles do not always burst. Geometry, 
as every one knows, is, etymologically, earth- 
measurement, and the corresponding term 



THE OLD THEOLOGY 13 

in the Chinese language means ^show it by a 
figure.' Geometricians know, however, that 
their science is not mainly concerned with 
measurement of any kind, much less with 
measurements of Earth, and they know that, 
far from depending on figures, which are 
things of sense and imagination, geometry 
is a purely conceptual architecture, always 
strictly and for the most part obviously 
transcending sense and imagination. The- 
ology may indeed, in the future as in the 
past, discourse about gods or about God; 
she may do so legitimately, conveniently, 
often consistently with an immense literature 
and a vast tradition. But to speak of "a 
science of God", even if the locution were 
clear, which it is not, could hardly serve as 
a felicitous indication of the subject-matter, 
or the field, of a science. It does not ring 
right: it sounds extravagant, pretentious, 
irreverent. In so far as God must be sup- 
posed to be superrational, the speech is 
absurd; and it has, moreover, the fatal 
disadvantage of seeming to exclude from the 
circle of theological thought the finest spir- 
itual meditations of many millions of our 
fellow men. If we insisted upon defining the 



14 THE NEW INFINITE AND 

province of theology etymologically, a 
devout adherent of so great and noble a 
religion as Buddhism, for example, however 
profound his understanding of spiritual 
things, would have to be denied, and he 
would disclaim, the interests and the charac- 
ter of theologian. Such a conception is 
shallow and narrow. The domain of what is 
to be called theology must be conceived with 
sufficient depth and catholicity to include the 
thought of all men and women, whatever their 
time, place or creed, whose vocation it is to 
cherish the kind of wisdom that seeks to 
understand and to interpret rationally the 
supreme ideals of the human spirit. 

Shall we, then, say that theology is the 
science of religion? There are, I think, 
insuperable objections to doing so. For one 
thing, the words have been gradually appro- 
priated in recent times to another use; they 
carry a different import ; they point to some- 
thing else. They point, on the one hand, to 
psychological analysis of the religious emo- 
tions and, on the other, to study of their 
external manifestations — to their sensible 
embodiments in institutions, customs, cere- 
monies, and rites. Such analysis and such 



THE OLD THEOLOGY 15 

study are important enterprises ; they are 
intimately related to theology ; in a measure, 
they fall within its scope, but only so as 
auxiliaries and adjuvants and not as con- 
stituting the center or bulk of its concern. 
Not only do they differ from theology in 
their attitude towards religion, being less 
warm, less sympathetic, less constructive, 
less philosophic in their interest and bearing, 
less interior and spiritual, but — what is more 
significant — they differ from it in respect of 
content and subject-matter. Theology may 
analyze the religious emotions, or try to do 
so, and it may study their exterior mani- 
festations in time and place, but these enter- 
prises are not, singly or jointly, its chief 
concern. Theology is neither a branch of 
analytic psychology nor a branch of anthro- 
pology nor yet a combination of them. What 
is called the science of religion — the anthro- 
pological study of religion — is related to 
religious life very much as botany would be 
related to the life of plants if we supposed 
plants to be conscious of what we call their 
life and if botanists were fairly repre- 
sentative vegetables. But before botany 
could develop a branch, or acquire an inter- 



16 THE NEW INFINITE AND 

est or a function, analogous to that of 
theology, it would be necessary to endow 
plant life much more highly than we have 
just supposed. We should have to suppose 
it endowed with fear and love, reverence and 
awe, hope and aspiration, with supernalizing 
power, with dreams and ideals, with respon- 
sive sensibility to the light of a higher 
world. The relation of theology to religion 
is, then, not that of a science to its subject- 
matter. Granted that religion is theology's 
source, its motive, reference, and goal, its 
raison d'etre. That does not mean that reli- 
gion is its subject-matter. We have seen 
that religion is essentially emotion ; theology 
is doctrine ; the former feels ; the latter 
thinks ; theology is a structure — an edifice 
of thought; religion is a flow — a stream of 
sentiment; theology is subject to the govern- 
ance of ideas, deriving its authority from 
the rules of reason; religion is under the 
sway of ideals, deriving its authority from 
reason's dreams; the materials of the former 
are near at hand, they belong to the domain 
of the rational; the emotions of the latter 
come from afar, having their ultimate source 
in a realm superrational ; the light of the- 



THE OLD THEOLOGY 17 

ology is the light of the understanding ; that 
of religion is the mystic radiance of an over- 
world. 

In this triune scheme of distinct but 
kindred things of the spirit, in this triple 
combination and interplay of idea, ideal, and 
feeling — of reason, overworld, and reli- 
gion — , it is now at length evident, I believe, 
where we are to find the province and the 
role of theology. It is evident that its part 
in the great drama is the part of idea and 
reason, the part of intellect. The subject- 
matter of theology is not immediately nor 
primarily the religious emotions nor is it the 
interior constitution of their superrational 
ground and source: it is neither the feelings 
themselves nor the essential inner nature of 
the overworld that thrills them into being 
and sustains their life. Its subject-matter 
proper consists of rational phenomena: it 
consists of those facts and processes of the 
rational understanding that serve at once to 
indicate the existence of an overworld and 
to manifest its shining upon the things below. 
It is thus the task of theology to study those 
implications of logical thought that are 
hyperlogical, and, in so far as possible, to 



18 THE NEW INFINITE AND 

interpret them in rational terms ; it is its 
function to examine the nature of rational 
thinking in its various categories, to unfold 
its hid intent, to clarify the manner in which 
thought, following endless courses within its 
own domain, perpetually approximates, for- 
ever pursues, and intimates, by the laws of 
its going, limits that lie beyond. There is 
in Reason a life-process deeper and finer than 
the mechanical movements of ratiocination, 
there is a kind of divine energy there, a 
beholding presence, a faculty within a fac- 
ulty, a soul, if you please, in Reason, that 
fills her heart with dreams, points to a shin- 
ing canopy above the summits of her thought, 
discerns in the light and atmosphere of her 
common activities the sheen of ideals — the 
glory of perfections — above and beyond 
them all. The nature and significance of 
that supernalizing power — there, I take it, 
is theology's problem. Theology is, in a 
word, the science of Idealization. 

It is a natural science ; not indeed a labor- 
atory science; not, in ordinary sense, an 
observational science, for the objects it ob- 
serves are inner things, things beheld only 
in psychic light, not things stained with 



THE OLD THEOLOGY 19 

refracted radiance of the sun; neither is it 
an experimental science save in the sense in 
which all thinking whatsoever — all logical 
procedure — is essentially experimental; but 
it is, none the less, a natural science. 
Granted that its materials are not things of 
sense; granted that they are things of 
reason — familiar shinings there of strange 
supernal lights ; they are not on that account 
unnatural. The phenomena of idealization 
are not artificial nor forced; they are spon- 
taneous, springing from foundations deeper 
than will; they are as natural as the dawn; 
their credentials are cosmic. 

What it is that makes the task of theology 
so difficult and delicate is clearly to be found 
in the peculiar character of its subject- 
matter — in the essential nature of it and 
especially in the relation it bears to the over- 
world. We have to do with the phenomena 
of idealization. There are no special diffi- 
culties to be encountered in dealing with the 
great process as a process; the major fact 
about it — that of its existence, its ubiquitous 
presence, its ceaseless operation — is plain; 
there is nothing insuperable in ascertaining 
where and how it begins — here, there and 



20 THE NEW INFINITE AND 

yonder, as we have seen — in every category 
of the rational understanding; nor in ascer- 
taining how it advances, from initial points 
in the domain of reason along innumerable 
paths of thought that run endlessly on and 
on, like an increasing sequence of terms, 
outward towards the border; nor yet in 
ascertaining how the process ends, in the 
presentation, namely, of limits that lie 
beyond. In all this, in all that pertains to 
the process as such, there are indeed difficul- 
ties, subtleties of thought, delicate considera- 
tions, but nothing of a kind to baffle the 
methods of science. But what shall we say 
of the results of the process, of the limits 
presented by it, of those great ideals them- 
selves of which it is the function of ideali- 
zation to make us aware .^ It must be noted 
and borne in mind that they are not con- 
cepts, they are not ideas, they are ideals. 
How is theology, how is theology as a 
science, to deal with them? Their similitudes 
and differences are to be detected; they are 
to be compared, ordered, and classified; 
their significance is to be appraised; their 
authority determined; their claim to su- 
premacy in the ascending scale of values 



THE OLD THEOLOGY 21 

must be examined. How may all this be 
done scientifically? In such an enterprise 
the primal instinct to seize and subjugate is 
of no avail. Ideals are not things to be 
grasped, they are things to be reached for; 
they are not subjects for conquest, they 
are objects for aspiration; they are not 
properties to be possessed, they are perfec- 
tions to be pursued; logic can not harness 
them, it can not reduce them, as it reduces 
ideas, to the ranks of obedient servants in 
the fields of reason; they hover aloft; they 
can not be pounced upon ; to realize an ideal 
is not to possess it ; it is to own its authority, 
to respond to its appeal, to follow its leading, 
to be drawn to higher elevations by the 
charm and persuasiveness of its majesty and 
beauty. It is evident, I believe, what must 
be the answer to the foregoing question. In 
dealing with the great ideals, theology must 
approach them from below, from their 
ground and source, which are a rational 
ground and source ; she must approach them 
through an understanding of the infinite 
sequences that have the ideals, not as final 
terms, in reason, but as superrational limits ; 
she can know them only as they are revealed 



22 THE NEW INFINITE AND 

in the mode and light of their genesis ; she 
must study them as results of a process — 
results that she can not immediately handle 
or seize — a process with which she is com- 
petent so to deal. 

An even greater source of theological 
difficulty and confusion is the subtle and 
bewildering relation the ideals in question 
bear to the overworld. Theology is to be 
rational, scientific. The overworld is super- 
rational. It is obvious that such a world 
can not be the subject of a science. It is 
evident that theology can not be a science 
of the overworld as astronomy, for example, 
is the science of the heavenly bodies, as 
physics is the science of matter and motion, 
as biology is the science of organic life, or 
as mathematics is the science of logical 
implication. To speak of explaining super- 
rational being in rational terms is folly. 
Does it, therefore, follow that theology 
must remain silent regarding superrational 
reality.^ It does not. The overworld has 
downward- facing aspects ; it presents aspects 
to the upward gaze of reason: these are 
reason's ideals, superrational limits, as we 
have seen, of rational thought. Of these 



THE OLD THEOLOGY 23 

theology may speak; she may speak of their 
origin, of the process and mode of their 
presentation, of their significance, of their 
majesty, of the lure of their beauty, of their 
glory; she may speak of their genuineness 
and authority, of their relation to hope and 
aspiration, yearning and love, reverence and 
awe. But she can not without folly under- 
take to explore nor pretend to explain the 
inner constitution, the ultimate nature, of 
an overworld. 

The task of theology, thus conceived, is 
one of exceeding delicacy. It is little wonder 
that in her long, long career she has often 
gone astray, that she has committed innu- 
merable blunders, that she has sometimes 
despaired, that she has frequently incurred, 
sometimes deservedly, the disrespect, the 
antipathy, even the contempt, of scientific 
men. It is little wonder, too, that she fares 
ill in a practician age, that she wins but 
little encouragement or support in compari- 
son with those physical sciences that have 
the advantage of being able constantly to 
vindicate their worth in the eyes of a tinker- 
ing and huxtering world through "useful" 
applications, multiplying the conveniences of 



24 THE NEW INFINITE AND 

men, advancing their physical welfare, ex- 
panding and subliming their petty pursuits 
to the proportions and elevation of vast and 
dazzling commercial and industrial enter- 
prises. There is no domain of thought, no 
branch of science or speculation, where the 
subject-matter is quite so subtle, where the 
facts are so intangible, so elusive, so remote 
from sound and touch and sight, where the 
conceptions are so tenuous, where the 
hypotheses are so generic and broad, so 
hard to verify, and where it is so difficult 
to discriminate appearance from reality, 
separating from out the wildering maze 
problems that are genuine from those that 
are not. It is precisely on this account, 
however, that modern mathematics, as I hope 
we may see, is qualified by the inmost char- 
acter of her being to lend a helping hand. 

The answer of Laplace to Napoleon's 
question, why he had not in his Mecanique 
Celeste mentioned the name of God, is known 
to all: "Sir," the savant replied, "I had no 
need of that hypothesis." Not so generally 
known is the instant response of the great 
author of the Mecanique Analytique when 
the Emperor made prompt report to him 



THE OLD THEOLOGY 25 

of the memorable conversation: "Neverthe- 
less," said Lagrange, "that is an hypothesis 
that accounts for many things." 

Let us not mistake the point of these fine 
words. Superficially the speeches appear to 
be mutually antagonistic ; they do somewhat 
resemble the sudden saber-thrust and counter 
thrust of battle. Yet they are in perfect 
accord. Their semblance of mutual oppo- 
sition is illusion, due to the dramatic char- 
acter of the situation and a certain contrast 
of sound. It entirely disappears on closer 
examination. There is neither irreverence in 
the one speech nor reverence in the other. If 
Laplace's mot indicate a lack of veneration, 
then that of Lagrange must indicate a lack 
of scientific temper. Scientific temper lack- 
ing in Lagrange ! It is true that Laplace, 
at the close of his immortal work, might, like 
Newton before him, have discharged the 
mood essential to its production; he might 
have given himself to another kind of medi- 
tation, to leisured contemplation of the 
cosmic visions gained in years of analytic 
toil; and thus receptively musing on the 
mighty mechanism of the stellar universe — 
its unfathomable deeps, the immeasurable 



26 THE NEW INFINITE AND 

energies of swift-revolving worlds of flame, 
the all-pervasive order, the silent reign 
throughout of majestic law — , he might 
have felt a reverent sense of admiration akin 
to religious awe, and — again like Newton — 
have owned in words that such unity and 
power betoken the dominion of a Supreme 
Ruler and Lord of all. Had he done so, had 
he thus chosen to crown his scientific work 
by some expression of belief in a divine 
source and ruler of a universe whose pro- 
founder beauties he had been enabled to 
behold and disclose, the testimony could not 
but seem fitting to everyone; it would be 
especially grateful to those fortunate folk 
who see in every great display of power a 
witness to omnipotence, in every striking 
manifestation of natural law an evidence of 
divine decree, in every nobler scene of beauty 
a token of divine perfection. But — and this 
is the thing to be noted — such an expression 
of belief, however creditable to the great 
astronomer in his character as a man, would 
not have been in any sense a constituent of 
the Mecanique Celeste — neither a postulate 
nor a theorem, no proper part whatever of 
the great description, but only an after- 



THE OLD THEOLOGY 27 

effect, a note of veneration evoked by subse- 
quent recall and contemplation of the celes- 
tial scenes described. Had some soldier of 
Euclid's time demanded of the illustrious 
geometrician why he had not in the Elements 
made mention of Zeus, no doubt the wit 
provoked but yesterday by the challenge of 
Napoleon's question had framed itself in 
Greek two thousand years before. Or does 
some one imagine that that least perishable 
work among the scientific monuments of the 
ancient world could have been scientifically 
improved by adding to its underlying postu- 
lates the statement, There is a God? If one 
asks, for example, why planetary paths are 
elliptic, or why camels have humps, or why 
the earth is flattened at the poles, and 
receives for answer that there is a God and 
that God so wills, the answer may indeed be 
a statement of fact, and yet as a scientific 
answer it would be absolutely worthless; it 
would be silly; and any one who could 
solemnly offer it as scientific would seem less 
logical than pathological. The resolute 
attempt of science to explain the universe 
in terms of mechanics can not be furthered 
by the postulation of a God ; indeed it would 



28 THE NEW INFINITE AND 

be abandoned thereby; for one thing is 
certain : God, if God there be, is no machine. 
Laplace was right; he had "no need of that 
hypothesis." Nay, his problem being one of 
mechanics, he could not, without stultifying 
himself, have even pretended to use it. 

"Sir, I had no need of that hypothesis." 
Laplace was right. "Nevertheless that is an 
hypothesis that accounts for many things." 
Lagrange was right. It is evident that the 
significance of the two speeches lies, not in 
their seeming discord, but in their real con- 
cord: in their common point of view; it 
consists in what neither one asserts but both 
of them imply: namely, that God is an 
hypothesis. 

Let me say, for what it may be worth, 
that personally I am far from prepared to 
contend that God is the name of an hypothe- 
sis and nothing more. It is perfectly true 
and perfectly clear that science, viewed as 
an attempt to explain, in mechanical terms, 
all phenomena, the attempt itself included, 
is, thoroughgoingly, an atheistic enterprise. 
It is a legitimate enterprise ; it is carried on 
under a working hypothesis that men may 
make — the hypothesis that mechanical prin- 



THE OLD THEOLOGY 29 

ciples are sufficient; under it great things 
have been achieved; there is every reason to 
expect that even greater things will follow 
with the years ; it is a right, it may be a 
duty, to pursue it for all it may yield. But, 
while Science, thus defined, is essentially 
atheistic, scientific Man is not. Man is 
greater, infinitely greater, than science, as 
he is greater than art or philosophy or 
religion or any mode or form in which his 
life may manifest itself. Many a scientific 
man is temperamentally disqualified to re- 
gard the mechanistic hypothesis as all- 
sufficient, and who is qualified to say that 
temperament has no essential relation to the 
problem? Many a scientific man, even the 
hardiest of the kind — unless cut off before 
the mellowing touch of pensive years can 
ripen knowledge into wisdom — comes sooner 
or later to feel that the mechanistic hypothe- 
sis, fruitful as it is, can not embrace the 
whole of life, that it can never give an ade- 
quate account of the finer elements of "man's 
unconquerable mind" — its radiance and joy, 
its conscience and love, its spiritual yearn- 
ings, its holy aspirations; and so, under the 
chastening influences of time and meditation. 



30 THE NEW INFINITE AND 

more and more awake to the subtler claims 
of his being, he comes, reluctantly perhaps, 
slowly it may be and late in life, to reconsider 
and rectify his earlier estimates, and from 
the doubt that is "hungry and barren and 
sharp as the sea," craves and seeks relief, 
finding it at length in a sense of a sympa- 
thizing consciousness not his own, in subtle 
intimations of the pervasive presence of a 
living Spirit. 

Neither do I deny that, far from being a 
mere hypothesis, God may be a real being — 
an infinite personality — whose reality is, at 
times, to persons of a certain temperament, 
an immediate object of a genuine kind of 
knowledge — the kind that mystics have 
sometimes claimed to have. That they have 
been sincere there is no reason to doubt. 
Have they been mistaken? I do not know. 
Knowledge of the kind in question is said 
to be ineffable. If it exists, it is ineffable. 
That does not mean that it does not exist; 
it merely means that, if it does exist, it is not 
scientific knowledge, for scientific knowledge 
is effable: it is communicable knowledge; it 
rests on a kind of evidence that, if it is for 
you, is also for me, or, if for me, then also 



THE OLD THEOLOGY 31 

for you — it is not essentially private or 
personal, it is essentially public and imper- 
sonal. But knowledge, we know, is not all 
of a kind. It would be stupid to maintain 
that all knowledge must be scientific or else 
ungenuine. I know how to move my arms, 
we say, or how to walk, to cast a stone, to 
wink, to swallow, or to think; a squirrel, we 
all say, knows how to climb a tree or gnaw 
a nut, a horse how to find its stall: such 
knowledge is not scientific. To my wife the 
full moon appears the size of a dinner plate ; 
to me, the size of a large cart-wheel. How 
big is the moon? That is not the question; 
if it were, the right answer would belong to 
scientific knowledge. How big does it seem.^ 
That is meaningless. How big does it seem 
to you? That you can know. To me? 
That I know. But your certitude and mine 
are not common to us — they are not imper- 
sonal, they are individual, private, personal 
certitudes. 

In this connection it is worth while to 
mention another type of evidence — a kind 
of evidence that is, like the mystic's, in- 
effable — at all events exceedingly hard to 
communicate — and yet is, I suspect, avail- 



32 THE NEW INFINITE AND 

able to the normal intellect, provided it will 
be at the pains to try a certain psychological 
experiment. The experiment relates to the 
great question of cosmic purposefulness. 
To deny the universe that quality is so easy 
to do in words. But to do so in fact — to 
gain, that is, a poignant sense of the denial's 
essential meaning — appears to be a matter 
of exceeding difficulty. May I refer to my 
own experience ? I have tried the experiment 
many times. Mood is essential, and time and 
place — springtime or autumn, evening or the 
still night, rural solitude under the moon 
and the stars. In the course of thirty years 
I have won, perhaps a hundred times, what 
seemed to be a realizing sense of what it is 
that the denial means. Words fail. To 
know the sense one must feel it. When it 
comes, it comes like a sudden apparition, but 
it does not stay. Its momentary presence 
seems to involve an instant's failure of its 
support, like a swooning of mind immediately 
checked and healed, like the integrity of 
being itself suddenly recovered from the 
brink of dissolution. The coming and going 
are quick as a streak of lightning; only the 
apparition is dark, like the passing shadow 



THE OLD THEOLOGY 33 

of a flitting bird, like a mid-day moment's 
dream of dusk at once dissolved in the light, 
like a cut in consciousness instantly closed 
as a cleft in a sea : the denial being no sooner 
achieved in feeling than it has been com- 
pletely overwhelmed by the inrushing flood 
of 'What, then, is it for?' — as if some sud- 
denly roused instinct, Tital to Intelligence, 
had leaped to the defense of her integrity 
and life. Such experience leads me to sus- 
pect that cosmic purposefulness is something 
profounder than a doctrinal postulate with 
which thought may, if it choose, dispense. 
I suspect it is an essential part of what mind 
means by mind. 

But, after all such claims have been duly 
allowed, we must not fail to see clearly that, 
for theology regarded as a purely scientific 
enterprise, God is an hypothesis and nothing 
more. For the rapt vision of the seer, faith's 
evidence of things not seen, the mystic's 
immediate sense of divine communion, the 
above-mentioned evidence of cosmic purpose- 
fulness, all these and their kind being essen- 
tially personal, private, ineffable, incommu- 
nicable experiences, are none of them forms 
of scientific knowledge: because, as I have 



34 THE NEW INFINITE AND 

said, scientific knowledge always is, poten- 
tially at least, impersonal, public, sharply 
discriminated in kind from other varieties of 
knowledge by what we may call its social 
character, by its transmissibility from mind 
to mind. Knowledge of the outward differ- 
ences between Greek architecture, for ex- 
ample, and Hindu architecture is scientific 
but your knowledge that one of these pleases 
you more than the other is not scientific, it 
is private. The idiosyncrasy of scientific 
knowledge is that, though perchance there 
may be at a given time but one individual 
who has it, yet it does not belong to him in 
his individual capacity ; it is his as a member 
of society, as a representative of humankind. 
Of expert logicians, for example, there may 
be in a given community but few. Yet the 
science is not theirs. Logic is impersonal. 
It belongs to Man. 

Here, then, we are face to face with a 
capital theme of theological meditation: the 
assumption, namely, or hypothesis of a being 
called God. How shall we frame it in speech? 
How describe the august Being it seeks to 
represent.? If we appeal to the greatest 
physical philosopher of all time, the author 



THE OLD THEOLOGY 35 

of the Principia and inventor of the Infinit- 
esimal Calculus returns the terse reply: 
"A Being eternal, infinite, absolutely per- 
fect." If we listen to him whose genius 
established the great alliance between the 
doctrines of Number and Space, thus bring- 
ing together the sundered hemispheres of 
apodictic thought and so creating the world 
of Analytic Geometry, we hear the resound- 
ing words of Descartes: "Infinite, eternal, 
immutable, independent, all-knowing, all- 
powerful." If we ask the "God-intoxicated" 
philosopher of Amsterdam, we receive from 
the great Spinoza a similar characterization 
not less impressive: "Absolutely infinite, 
consisting of infinite attributes, each express- 
ing eternal and infinite essentiality." These 
familiar citations will serve to remind the 
reader of like efforts, among the best of 
human thought, to formulate in adequate 
terms the hypothesis God. About things 
that are very familiar it is exceedingly 
difficult to bring ourselves to think, and the 
terms of the hypothesis have been familiar 
for hundreds of years. Were it new and 
fresh instead of being so old and stale, we 
should all of us be immediately struck by 



36 THE NEW INFINITE AND 

what is, among its distinctive features, a 
very obvious mark. The hypotheses that we 
meet elsewhere, as the nebular, the corpus- 
cular, the ionic, the atomic, the molecular, 
the hypothesis of a space-pervading ether, 
of universal gravitation, of Euclidean 
space, of organic evolution, of conservation 
of energy or of mass, all such — all the 
hypotheses we encounter in the literature of 
ordinary science — have one character in 
common : each of them is restricted in scope, 
limited to some fragment of reality, they 
divide in order to conquer, each is confined 
to a field that is bounded; not so, however, 
the hypothesis God: it is distinguished by 
the fact that, among hypotheses, it alone 
attempts to span and bind the Whole. That 
is a very remarkable characteristic. And 
soon we must note another. But first we 
must ask, What does the hypothesis mean.^ 

"The light of human minds," says Hobbs, 
"is perspicuous words, but by definitions 
first snuffed and purged from ambiguity." 
Accordingly it is necessary to ask: what, if 
any, precise meaning, available for the pur- 
poses of logical discourse, may be assigned 
to the terms of the hypothesis? Infinite, 



THE OLD THEOLOGY 87 

Eternal, Omnipotent, Omniscient, Omni- 
present, and the rest : what do these mighty 
terms mean? I do not now ask for their 
meaning as instruments for energizing life. 
I do not now ask for their meaning as cries 
of the spirit — voices from the deeps of feel- 
ing. At present I am not concerned with 
their meaning for reverence, for love, for 
awe. I do not here seek their relation to the 
moods of poetry and prayer. I am not en- 
quiring for their emotional significance, so 
like that of mountain scenery, a vast wilder- 
ness, the heavens above, or the "solemn 
anthem of the sea." I enquire for their 
logical value, for their meaning in Thought. 
It is essential to note at once a very remark- 
able thing: the great terms in question are 
not names of scientific notions, they are not 
names of concepts, they are not names of 
ideas ; they are names of ideals — super- 
rational ideals, outlying limits of rational 
thought. This gives the hypothesis an ap- 
pearance of being an hypothesis respecting 
the nature of the overworld. Is it such in 
fact? So to take it, as many consciously or 
unconsciously do, is fatal. It removes the 
question from the jurisdiction of theology 



38 THE NEW INFINITE AND 

regarded as a science: a world of super- 
rational reality can not be the subject of 
any science; to suppose the contrary is to 
ignore the sole restriction that the muses 
have placed on freedom of thought: thought 
must be free from internal contradiction, it 
must be harmonious. It is not necessary, 
however, to take the hypothesis so. The 
overworld, we have seen, has downward- 
facing aspects. These aspects, presented to 
the upward gaze of reason by the process 
of idealization operating in reason's fields, 
are precisely the ideals that the terms of our 
hypothesis serve to designate. The hypothe- 
sis is accordingly to be regarded as an 
hypothesis respecting, not the essential inner 
nature of the overworld, but the nature of 
the downward-facing aspects presented by it 
through the process of idealization — a super- 
nalizing agency working below. But, if the 
ideals, the aspects in question, are super- 
rational, how is it possible for science to say 
aught about them.^ Science, we have seen, 
must approach them from below, in the light 
of their genesis and manner of presentation. 
By this method science is enabled to say of 
them that such and such ideals are of a 



THE OLD THEOLOGY 39 

nature to appear as limits of such and such 
processes or sequences of rational thought. 
To be able to say that, however, with all it 
implies, is much: just as it is much — if I 
may illustrate* great things by small — to be 
able to say of a curve, for example, that, 
though outside the domain of broken lines, 
it is yet the limit of an endless sequence of 
broken lines ; or just as it is much to be able 
to contemplate a curved surface as an ideal 
indicated by an endless series of plane- 
bounded figures approximating it forever, 
though the ideal itself does not belong to the 
field of the approximating figures; or just 
as it is much to be able to view what is called 
an irrational number as an ideal or limit 
beyond the domain of rational numbers but 
indicated and endlessly pursued by series of 
these; or just as, in general, it is important 
for the life of understanding, to be able to 
make out, in whatever field it operates, the 
endless courses of ever increasing approxi- 
mation that by the law of their progress at 
once betray appropriate perfections beyond 

* For a fuller explanation of the force and point 
of such illustrations, see the author's "Science and 
Religion," herein cited on an earlier page. 



40 THE NEW INFINITE AND 

and, though never attaining them, yet lead 
us more and more deeply into their far- 
shining light. 

Such, then, must be the method and such 
the ways of rational theology. If it is to 
have a motto, the motto must be: From 
ideas to ideals. The former indicate, the 
latter are indicated. These are to be under- 
stood scientifically only in so far as their 
meaning is revealed in the ideas indicating 
them and in the manner of the indication. 
Among the ideals with which we are here 
concerned, among the great ideals assembled 
in the hypothesis God, it is obvious that there 
is one which has the distinction of seeming 
to be at once coordinate with the rest and 
yet in a sense involved in each of them. I 
refer, of course, to the ideal denoted by 
the adjective Infinite. There is a Being, 
so the hypothesis runs, at once infinite and 
omniscient and omnipotent and, so on. 
Omniscience, however, involves Infinitude ; 
so does Omnipotence; so does Eternality; 
so does every pealing note of the great 
diapason. Any illumination of what is 
meant by the term Infinite will, therefore, 



THE OLD THEOLOGY 41 

serve in a measure to illuminate the meaning 
of the kindred terms. 

The reader doubtless knows that the 
Infinite of theology has never been defined — 
defined, that is, for logical use — , and he is 
now in a position, I believe, to see why it has 
not. It is not because the centuries have not 
witnessed many ingenious attempts to define 
it. It is because the term denotes, not an 
idea, but an ideal, a superrational ideal, and 
so does not admit of definition. There can 
be no doubt that a great deal of the confusion 
found in theological literature has resulted 
from the fact that theologians, failing in 
respect of this logical distinction, have gone 
on discoursing about what they have called 
the Infinite, as if the term stood for some- 
thing — a concept or an idea — that had been, 
or, at all events could be, defined. The 
remedy for the kind of confusion that thus 
results is simple: it consists in not ignoring 
the distinction; it consists in ceasing, once 
for all, the attempt to treat an ideal as an 
idea ; it consists in refraining from the hope- 
less endeavor to deal with a superrational 
limit as if it were a rational term of the 
endless sequence whose nature it is, not to 



42 THE NEW INFINITE AND 

contain the limit nor to attain it, but to 
indicate it and approximate it. 

There are, however, other difficulties con- 
nected with the term — difficulties inherent 
in the nature of the case, proper difficulties, 
we may say, because they belong to the ideas 
constituting what we may call the ideal's 
rational basis, its basis in reason. Here it 
is necessary to note an important distinction, 
to be henceforth kept in mind. Thus far we 
have been speaking of the theological infinite. 
Fortunately or unfortunately the same term 
is constantly employed in science and espe- 
cially in mathematics in another sense — in 
a sense closely related indeed, as we shall 
see, to theology's sense of the term but yet 
quite distinct therefrom. In theology, we 
have seen, the term denotes an ideal, a super- 
rational ideal, which can not be defined; in 
mathematics, as we are going to see, it de- 
notes an idea, a concept that not only is 
sharply definable but has been in fact sharply 
defined. Presently we shall begin to see the 
beautiful relation between the two senses in 
which the term is employed and why it is 
and wherein the mathematical sense is an 
indispensable means for making clear the 



THE OLD THEOLOGY 43 

theological sense. The ideas — the objects 
or things — that mathematics calls infinite 
are not all of them of one order. There are 
countless mathematical infinities, or infini- 
tudes, or infinites, as they are variously 
called, — countless types of them. Like the 
stars, they differ in glory. They constitute, 
as we shall see, an endless sequence of ever 
increasing terms — an endless series or suc- 
cession of terms mounting ever higher and 
higher in respect of order or dignity or rank. 
Each term in the endless march of terms 
includes the type of infinitude represented 
by the preceding term but is itself of higher 
type. The major relation in the scheme 
delineated is evident at once : the limit of this 
endless series of infinite ideas is an infinite 
ideal: the Infinite of theology is the limit of 
the endless sequence of more and more 
embracing Infinitudes presented by science. 
It is not one of them ; it is, so to speak, their 
envelope, enfolding them all. 

It is now time to look into the great rela- 
tion a little more deeply. We must try to 
see quite clearly what scientific infinities are ; 
we must endeavor to understand how they 
are related to the finite things of sense and 



44 THE NEW INFINITE AND 

how they embrace and penetrate the common 
affairs of men; we must learn something of 
the law in accordance with which they are 
disposed, rank above rank, in a hierarchy of 
orders, without a summit; we must observe 
how the process of Idealization, operating 
in and among them, pervades the atmosphere 
of the grand array, and creates or finds there 
a subtle radiance that seems to reveal a down- 
ward-shining aspect of an overworld. The 
task is not an easy one; it demands a little 
patience and a little penetration; a part of 
the discussion, which is for such as prefer 
not being entertained to being fooled, must 
seem to some a little arid — a pretty dry way 
to a valley of fruits ; a sense of its full 
significance can not be gained at once; it 
must be won as the fruit of reflection. 

In order to explain the scientific or 
mathematical meaning of the term infinitude, 
or infinite, let me begin with some simple 
examples. I will take them from the two 
hemispheres of rigorous thought, the two 
great subject-matters of it — Number and 
Space. 

Imagine two concentric spheres, the inner 
one white and named the silver sphere, the 



THE OLD THEOLOGY 45 

outer (or larger) one yellow and named the 
golden sphere. (In accordance with the 
usage of higher geometry I shall mean by 
sphere a sphere-surface.) Next imagine the 
sheaf (as it is called) of rays, consisting of 
all the straight lines that have their begin- 
ning at the common center of the two spheres 
and thence extend outward endlessly in every 
direction. It is plain that any ray, 1?, of the 
sheaf pierces the silver sphere in a point, 
say S, and the golden sphere in a point, say 
G, Calling S and G a pair of points, it is 
evident that, by considering all the rays of 
the sheaf, the points of the one sphere are 
paired with those of the other in a one-to- 
one, or point-to-point, fashion: in other 
words, a unique and reciprocal correspond- 
ence is thus established between the points 
of the silver sphere and those of the golden 
sphere. One silver point corresponds to one 
and but one golden point ; one golden point, 
to one and but one silver point; and this 
reciprocal relation holds for every silver 
point and for every golden one. We see at 
once that the number of points on the one 
sphere is exactly the same as the number 
of points on the other ; we see, too, that this 



46 THE NEW INFINITE AND 

number equality subsists no matter how great 
the difference between the sizes of the 
spheres — one of them may as well be micro- 
spically small and the other billions of times 
larger than the earth or the sun. In other 
words, the number of points on a surface of 
given size (given area) is independent of the 
given area or size, and so will not be changed 
by changing the area or size. Now imagine 
a closed curve or ring — red, if you like, for 
the sake of vividness — to be drawn on the 
golden sphere and enclosing thereon a por- 
tion of it, a region A, precisely equal in area 
to the area of the silver sphere. We need 
not suppose this latter equality (of areas) 
but we may as well, for the supposition will 
reduce a little the shock we are soon to 
receive. The number of points in the region 
A is, of course, the same as the number on 
the silver sphere and is, therefore, the same 
as the number on the golden one. But the 
collection of points in the region A is only 
a part of the whole collection on the golden 
sphere. The shocking thing is this : we have 
here a part — the ensemble of points in the 
region A — and a whole — the ensemble of 
points on the golden sphere — such that the 



THE OLD THEOLOGY 47 

number of points in the part is precisely the 
same as the number of points in the whole. 
It is to be noted carefully and once for all 
that the astonishing equality subsists, not 
between the area of the region A and that 
of the golden sphere, but between two multi- 
tudes (of points), of which one is a part and 
the other the whole. 

Does some non-mathematical reader, un- 
familiar with this kind of thinking, distrust 
the argument, feeling perhaps that he has 
been tricked by a juggling use of the notions 
of surface area and point collection? If so, 
let him scrutinize the equivalent following 
argument, in which the notion of area plays 
no role. First, a preliminary word of expla- 
nation. If a straight line and a plane are 
parallel, we say sometimes — in high school, 
for example, — that they have no common 
point ; but if we continue our study into what 
is called projective geometry, — never mind 
the name — , we learn to say that, if a line 
and a plane be parallel, they have a common 
point — called an "ideal" point to distinguish 
it from ordinary points — the "ideal" point 
being so far away that it can not be reached 
by a step-by-step process of going towards 



48 THE NEW INFINITE AND 

it; any "ordinary" point can be so reached. 
Such "ideal" points of a plane make up a 
line, called the "ideal" line of the plane. 
A plane thus conceived as having such an 
"ideal" line is called a projective plane, and 
a line regarded as having an "ideal" point 
is called a projective line. And now the 
promised argument. Think of a hemisphere, 
H, and suppose it to rest on a horizontal 
plane, /7, the hollow of H being open to the 
upper sky. The rim of iT is a circle, C. 
Denote its center by P. There is a sheaf 
of rays running out from P. Of this sheaf 
consider only those rays that lie in the plane 
containing C and those that run below this 
plane. The imagery is perfectly clear. The 
rays considered, since each of them pierces 
fl^ in a point and 27 in a point, plainly estab- 
lish a one-to-one correspondence between the 
points on the hemisphere H and the points 
of the plane II; a point on the rim of H 
obviously corresponding to an "ideal" point 
of 77, and conversely. Now imagine a plane 
above 77, parallel to it, and cutting H in two. 
Cast away the upper part of H so cut off 
and keep the lower part — the up-turned cap 
resting on II as before. The rim of this cap 



THE OLD THEOLOGY 49 

is, again, a circle. Call it C' and its center 
P', As before, there is a sheaf of rays run- 
ning out from P' . Of this sheaf consider, 
as before, only those rays that lie in the 
plane of the cap's rim and those that run 
below this plane. Again the situation is 
clear: the rays considered, each piercing a 
point from the cap and a point from 77, set 
up a one-to-one correspondence between the 
points of the cap and those of 77. We now 
see that the number of points of H is the 
same as the number of points of 77 and that 
the number on the cap is the same as the 
number on 77; hence, we see, the number on 
H is the same as the number on the cap. 
But the collection of points on the cap is a 
part of the whole collection on 77. The fact 
is, accordingly, now perfectly evident — 
whether at first we like it or not — that we 
are in a world where it is easy to encounter 
a whole having a part whose elements are 
precisely as numerous as are the elements 
of the whole. Every whole of that kind is 
said to be infinite. 

Be it understood, then, that the concept 
of infinity — the scientific or mathematical 
meaning of the term — is this: namely, a 



60 THE NEW INFINITE AND 

collection, class, set, group, aggregate, 
ensemble, manifold, or multitude of elements 
or things — be these points or passions, ions 
or ideas, relations or terms, quantities or 
qualities, numbers or instants or colors or 
sounds, degrees of wisdom or goodness or 
power or joy, or any other modes, forms, 
or determinations of being — is said to be 
infinite if and only if the collection, like the 
ensemble of points on a sphere, contains a 
part, or subcoUection, that is numerically 
equal to the whole. On the other hand, a 
collection or multitude is said to be finite 
if and only if, like the collection of trees in 
yonder forest, like the human population of 
the globe, like the multitude of sands of the 
sea or that of the stars within telescopic 
range, it contains no part or subcoUection 
numerically equal to the whole. 

There is here no ground for quibbling, 
hesitance, or doubt. There stand the two 
concepts, absolutely clear; and there, too, 
stand the validating facts, absolutely unmis- 
takable. The latter indeed may be multi- 
plied at will. Examples of collections 
illustrating the concept of finitude are of 
course familiar to every one, being forced 



THE OLD THEOLOGY 51 

upon the attention by the vulgar necessities 
of life; indeed they are so familiar that but 
few persons have so much as dreamed that 
there are collections or manifolds of another 
type. We have seen, however, that there are, 
and the gain is one — if we really make it our 
own — to work a profound transformation in 
our view of the world. Of examples illus- 
trating the concept of infinitude, we have 
thus far instanced but two. Similar examples 
abound, however, in even greater profusion 
than the other kind, being found in the great 
and the small, the remote and the near, in 
Number, in Space, in Time, in qualitative 
distinctions, in the realm of pure relation — 
wherever the intellect may penetrate — if the 
inner eye be only disciplined to detect their 
omnipresence. A little patience, I have said, 
is indispensable in this part of the discus- 
sion — quite as needful as a little penetration ; 
and I must request the reader's permission 
to tarry yet a little in order to point out a 
few further illustrations of what science 
means by an infinite multitude. It is of the 
nature of doctrine to grow aloft, higher and 
higher, into the limpid atmosphere of pure 
theory, and that is legitimate — architecture 



52 THE NEW INFINITE AND 

must rise ; but, however high its head, a doc- 
trine, if it is to stand, must plant its feet 
upon the solid earth of fact. The facts with 
which we are here concerned are not facts of 
sense; they are facts of thought; they do 
not belong to the domain that we, as animals, 
share jointly with the beasts; they are the 
prerogatives of man as man — in his capacity, 
that is, for "discourse of reason." Let us 
return for a moment to our image of the 
sheaf and the concentric spheres. Consider 
those rays of the sheaf that pierce the points 
of the region A on the golden sphere. Let 
us call the bunch of these rays a bundle. 
It is evident that the number of rays of the 
bundle is the same as the number of points 
in the region A, one ray through each 
point of A, one point of A on each ray of the 
bundle: this number, we have seen, is the 
same as the number of points on the sphere ; 
and this, again, the same as the number of 
rays of the entire sheaf; whence it is seen 
that the bundle, though but a part of the 
sheaf, has the same number of rays as the 
number of rays in the whole. And so the 
sheaf and the bundle serve to exemplify 
again the concept of infinite manifolds. 



THE OLD THEOLOGY 63 

Let me now take a very simple example 
from the inexhaustible resources of another 
field. Consider the little equation, y = 2^, 
which every one understands. If we assign 
a value, say 1, to «r, then, as we see, the value 
of 2/ is thereby also determined: it is just 
twice as much — in this case 2; if we let 
^ be ^, 2/ must be 1 ; if ^ be V2, y is 2 V2 ; 
and so on: to any value of the variable ^, 
there corresponds one and but one value of 
the variable y; and conversely, for we could 
just as well give values to y and so find for 
each of them its half, or the corresponding 
value of X. Let us now agree to let «r vary 
in value from zero to 1, taking, one at a time, 
the value zero, the value 1, and each of the 
innumerable host of values between; then y 
will take, one at a time, each of the values 
in the range from zero to 2, including, of 
course, zero and 2. Thus is set up a one-to- 
one correspondence between the multitude 
of values in the ^-range from zero to 1 and 
the multitude in the ^/-range from zero to 2. 
The number of values or numbers in the one 
range is, therefore, the same as the number 
of values or numbers in the other. But the 
collection of numbers in the range from zero 



54 THE NEW INFINITE AND 

to 1 is but a part of the whole multitude in 
the range from zero to 2. Accordingly each 
of these multitudes is an infinite multitude 
of things. 

As a final example here, let me invite care- 
ful attention to an infinite collection that 
would be the easiest of all to grasp were it 
not so very simple and if action upon it 
of the higher understanding were not almost 
inhibited by our fixed habit of regarding the 
elements of the collection as having no sig- 
nificance beyond their familiar vulgar uses 
in the counting-house and the market place. 
For the elements in question are nothing 
more romantic than the numbers with which 
we count. How very prosaic the prospect, 
you naturally say. I quite agree, and yet I 
venture to say that, if we will but rise above 
the stale levels of sense and imagination, we 
shall not fail to detect here a species of genu- 
ine poesy — the poesy of pure thought in 
touch with the infinite and eternal. Consider 
the two sequences or series of integers : 



(TT) 1,2,3,4,5,6, . . . . ,n, n + 1, . . 
(P) 2, 4, 6, 8, 10, 12, . . . . , 271, 2(n + 1) 



By the series {W) of symbols I wish to call 
attention, not to that uncompleted row of 



THE OLD THEOLOGY 66 

marks itself, but to a certain definite invisible 
whole that the row suggests and serves to 
bring as an object before the mind, namely: 
the totality of the positive integers. On 
being confronted with the notion of this 
fundamental totality, at once so clear to 
thought and so baffling to imagination, many 
persons, especially the uninitiated, become 
restive for a time. A little reflection, how- 
ever, will dissipate any reasonable scepticism, 
and show that our footing here is solid rock. 
It is true indeed that, however many integers 
we may singh' specify or imagine, there 
always remain more and more. It is also 
true that the hand cannot actually write nor 
the physical eye behold a set of symbols 
matching one-to-one all the integers com- 
posing the asserted totality, if such a thing 
there be. What of it? Consider, for a 
moment, a familiar totality so obvious that 
none may question it — the totality, I mean, 
of the points of a circle. As in the case of 
the integers, so here, too, it is impossible to 
think all the points singly or singly to specify 
or symbolize them all. Yet there they are — 
not one now and then another — but all of 
them at once, a totality persisting as such 



56 THE NEW INFINITE AND 

and unescapable. What is the secret? The 
secret is that the totality is a conceptual 
thing, a thing for thought and not for sense 
or imagination, a thing carved out by a law 
transcending the powers of step-by-step 
perception or depiction, a law of definition 
that selects out of the universe of thinkable 
things a set of them unambiguously — the 
law, namely, that the things shall be points 
of a plane and be all of them equally distant 
from a point therein. So it is precisely with 
the totality of positive integers. If you say 
that the totality does not exist, what you 
mean is that the integers of such a totality 
can not be written down for sight to look 
at or that no one can depict them all on the 
canvas of imagination. Permit me to remind 
you that I am not here addressing your 
sense nor your imagination. I am addressing 
your conception, your thought. The asserted 
totality does not exist for sense, it does not 
exist for imagination; it exists for thought. 
It derives its completeness and one-ness from 
the completeness and one-ness of the selective 
law defining it — the law, namely, that after 
any definite integer there is another greater 
than that hy one. Note that the law includes 



THE OLD THEOLOGY 67 

and excludes and that the inclusion and 
exclusion are both of them precise, decisive, 
complete, and instantaneous. It is pathetic 
if one can not see clearly that it is precisely 
such sense-transcending and imagination- 
transcending totalities that constitute the 
essential subject-matter of rigorous thought. 
For to deny their validity is to evacuate the 
Reason of its proper content and to bar even 
the possibility of Science. Science, properly 
speaking, does not deal with a set of things 
that we might fancy arranged in a row, like 
a row of blocks, beginning here and ending 
there. Science is interested only when the 
row, if it begins, never ends. Consider a 
curve. You can not exhaust its points by 
naming one after another of them. That 
is just why science is interested. It deals 
with the totality of the points by dealing 
with the curve, that is with the law — a 
definite thing — defining the totality, which 
is, therefore, also definite. Do you think 
geometry would exist if the points of space 
could be counted like a heap of marbles? If 
it did, it would be trivial. So much by way 
of reassuring those timid persons who, pri- 
marily children of sense and imagination, are 



58 THE NEW INFINITE AND 

filled with doubt and trepidation when asked 
to pass upward from their accustomed atmos- 
phere into the ether of pure thought. 

Let us now resume the advance. Compare 
the totality ( W) of integers with the totality 
(P) of even integers. Let us agree to pair 
each integer of ( W) with the one below it in 
(P). In this way a one-to-one correspond- 
ence is set up between the integers in (W) 
and those in (P), a result that we may indi- 
cate by the following sequence of pairs : 

(T) 1,3;2,4;3,6; ;n,2n; . . . 

Observe that the pairing is no creeping 
performance that never gets performed — 
ever going on and never finishing; neither is 
it a lightning-swift process, for this were 
as helpless before the task of pairing the 
totalities step by step as would be the pace 
of a snail: an endless course can not be run 
through by merely going fast. No, the 
pairing is an instantaneous deed of law, 
wrought without lapse of time. The law is : 
each number shall go with its double. To 
choose the law is to say : Let the pairing be 
done; and — it is done. To contemplate the 
deed requires time ; but the doing of it, none. 



THE OLD THEOLOGY 59 

There is possible a yet deeper view of the 
matter. I mean the static view. We may 
say, that is, — and this is correct — , that the 
integers as elements of the existing world of 
ideas already and always stand at once in 
all sorts of interrelations of which it is the 
nature of integers to admit, among such 
relations being that indicated by (T). In 
this view, the pairing is not a process of 
associating an integer with its double, then 
another with its double, and so on, thus 
establishing progressively, so to speak, the 
relation (T). It is not that; it is simply 
a single act of will choosing out a certain 
eternal relation from among hosts of rela- 
tions also eternal. Whichever view of the 
matter be taken — and either is admissible — 
it is clear that a one-to-one relation does 
subsist between the elements in (TF) and the 
elements in (P). The two totalities are 
therefore equally rich in elements: the 
number of integers in the one is the same 
as the number of those in the other. But 
every integer in (P) is an integer in (TF), 
while {W) has integers that are not in (P). 
Hence (P) is a part and {W) the whole; 
and so {W) contains an infinitude of inte- 



60 THE NEW INFINITE AND 

gers; and the like is true of (P), for what- 
ever matches an infinite, in the way now 
repeatedly exemplified, is, of course, itself 
infinite — indeed, infinite of the same rank. 

It is needless, I trust, to cite here further 
examples. "These slight footprints suffice 
to enable a keen-searching mind to find 
out" — not ''all the rest", as the maddened 
poet sang — but more and more. For, to 
eyes once opened, the brood of the infinite 
is everywhere. The light of the great con- 
cept shines in every aspect of being. The 
reader is now aware that this our world is 
a world that presents two great types of 
wholes, or manifolds, of thinkable realities — 
manifolds that are finite and manifolds that 
are infinite. He is now aware that each of 
the latter is characterized by the marvelous 
fact that it is a whole containing a part 
(countless parts indeed) matching the whole 
perfectly, as we have seen, in elemental 
wealth, in richness of content, in dignity of 
structure. The principle of discrimination 
is very simple — so simple indeed as to have 
eluded the eye of thought for thousands of 
years — for the doctrine is very^ modern, a 
faint first glimmer of it appearing in a work 



THE OLD THEOLOGY 61 

of Galileo and a little later in a hint of 
Pascal but not again, it seems, for two cen- 
turies. By it the universe of thinkable 
reality, as we now see, is riven asunder, not 
spatially indeed but logically. The two 
grand divisions — the realm of the finite and 
the realm of the infinite — , which are wonder- 
fully interlocked, together constitute a dual 
world answering to our dual life, the life of 
action and the life of thought. The realm 
of finite things is the domain of action, of 
Practical Life : it contains no multitudes but 
man may count them — the coins in the coffer, 
the cattle in the field, the deeds of a hero, the 
years of an empire; any series in it begins 
and ends; no totality or whole found there 
is matched by one of its parts : the world of 
finite things is an island-world suspent in a 
sea. And what is the immersing sea.'^ It is 
the realm of infinite things — an ocean with- 
out bottom or surface or shore. It contains 
no totalities but such as are law-defined, 
never a whole of any kind that has not 
countless parts each matching it perfectly 
in respect of number of elements, coequal 
with it in Mdchtigkeit as it is called, in 
potence or power, in complexity of structure, 



62 THE NEW INFINITE AND 

in dignity and wealth of reality. This is 
not the domain of Practical Life, though it 
penetrates the latter domain, intersects it 
in numberless ways, surrounds it, contains 
it in a sense: for a series that terminates is 
but part of one that does not ; every ensemble 
that admits of tabulation is a fragment of 
one than can not be fully represented by 
tabulation but only by a law; every whole 
that is an overmatch for its every part 
belongs to some vaster whole owning parts 
with respect to which it is not an over- 
match; every finite manifold is a sub- 
collection of an infinite one. No, the realm 
of infinite totalities, though embracing, in 
the sense explained, the domain of Practical 
Life, is not that domain; it is the domain of 
Reason, the province of Thought, the realm 
of Science; for, as Poincare has acutely 
pointed out, there can be no science, prop- 
erly speaking, of a finite subject-matter. 

Very well, one may wish to say, I grant 
what you have said, but what of it.? Where, 
pray, is Deity .^^ I ask for bread; you give 
me a stone. I ask for a vision of God; you 
invite me to thread endless mazes of mathe- 
matics; you invite me to contemplate vast 



THE OLD THEOLOGY 63 

and dazzling splendors of Number and 
Space. What does it all avail? 

" I heap up numbers enormous, 
Mountains of millions extend, 
Piling time upon time, 
World on world without end, 
But when from the awful height 
I would a vision of Thee behold: 
The total sum of number's Might, 
The' multiplied a millionfold. 
Is yet no part of Thee/' 

The protest is easy to understand, it is 
temperamental. May I reply, by way of a 
reminder, that I have promised no "vision'' 
of God? I am dealing with the hypothesis 
God. My aim is to throw some light on the 
meaning of its mighty terms. Chief among 
these is the theological Infinite. In pur- 
suance of the aim I have been here trying 
to clarify the meaning of scientific infini- 
tudes, of which the Infinite of theology is 
the supernal ideal or limit. None but the 
infinite, it is said, can comprehend the infi- 
nite. How familiar are the words ! How 
often have they been solemnly pronounced 
in courts of philosophy and sunken in the 
soul like a leaden decree of fate! But are 
they not true? "Comprehend" here means, 



64 THE NEW INFINITE AND 

of course, comprehend rationally, it signifies 
to understand as ideas are understood. Said 
of the infinites of science, the words are, 
then, true. Said, however, of the theological 
Infinite, they are neither true nor false ; they 
are meaningless, for the theological Infinite 
is, as already said and as I hope we are 
beginning to see, a superrational ideal, and 
to talk of comprehending or not comprehend- 
ing such an ideal as we talk of understanding 
ideas is not to utter what is true or false 
but what is void of meaning. If, however, 
^comprehend the Infinite' be — contrary to 
usage — taken to mean, not comprehend it as 
an idea, which it is not, but to signify having 
or gaining that kind of sense of what it 
means which comes from regarding it as the 
ideal or limit of infinites that we can com- 
prehend as ideas, then the old maxim, 
applied to the theological Infinite, is false, 
for we can win the mentioned sense in the 
mentioned way, and we do not, I believe, 
regard ourselves as superrational ideals. 
But have we not involved ourselves in con- 
tradiction.? For how can we gain the men- 
tioned sense in the mentioned way, seeing 
that this way requires ordinary, or logical, 



THE OLD THEOLOGY 65 

comprehension of infinites, and seeing that, 
regarding these infinites of science, we have 
admitted that none but the infinite can 
comprehend the infinite? The answer is no, 
there is no contradiction, for we are our- 
selves infinite in the scientific meaning of the 
term, where by "we" I mean the common- 
wealth of ideas over which your mind or mine 
can range. That this is true we know at 
length, thanks to mathesis, and we know it, 
not merely as an intimation or intuitive 
apprehension, but as a proved proposition 
of science. We know, that is, as Richard 
Dedekind has rigorously demonstrated, that 
the world of man's ideas as ideas — the human 
Gedankenwelt as the author calls it — is an 
infinite manifold. Shorn of contest and 
other non-essentials, the proof may be ren- 
dered in a line. Some readers will not 
require it. A friend tells me that he does 
not understand the proof and does not need 
it. I may add that he is a man of extraor- 
dinary spiritual sensibility. Intuition, how- 
ever, precious as it is, is often wrong, and 
I give the proof for the comfort of those who 
think it important to submit their intuitions. 



66 THE NEW INFINITE AND 

when it is possible, to the rigors of logical 
demonstration. 

Denote by G the Gedankenwelt — the world 
of ideas ; by / any idea therein, as that of a 
meal, a song, a deed of charity, a bargain, 
a moon beam, a diamond, a birth, a death; 
by 7i the idea of 7, for plainly the idea that 
a given idea is an idea is another idea; by 
I2 the idea of /i; and, generally, by /n+i the 
idea of /n. As any thought may itself be 
object of another thought, as this one may 
be object of a third different from the former 
two, and so on forever, it is seen that 7n+i 
can never fail, however large n may come 
to be, and so we have the two totalities : 

(TJ /, /l, /2, . . . . , /n, /n+l, . . . . ; 

(T') h, /2, /3, . . . . , /n+l, /n+2, . . . . ; 

the latter is a part of the former; each of 
them is a part of G. Now pair the two 
totalities as in the following scheme: 

/, /i ; /i, I2 ; I2, /s ; . . . . ; /n, /n+i ; /n+i, /n-1-2 ;....; 

each thing in (T) being thus associated with 
the thing below it in (jT'). At once it is 
seen that the whole totality (T) is com- 
pletely matched in one-to-one fashion by its 



THE OLD THEOLOGY 67 

part (T'); whence it follows that (T) is 
infinite; that {T) is infinite; and, a -fortiori^ 
that their common container, G, the Gedan- 
kenwelty is infinite. 

In this simple demonstration, so free from 
pomp, and in its conclusion, so significant 
for a right conception of man, there is large 
gain for rational theology, if indeed we may- 
hope that professional theologians will one 
day be moved to avail themselves of such 
considerations. It is no small gain to vindi- 
cate by logic a great intuition of the soul: 
it is no small thing to know, not merely at 
times to feel, that our faculties are framed 
to comprehend, scientifically, infinite elements 
in the architecture of the world. For in the 
presence of such knowledge, the terrors of 
Naturalism dwindle and vanish. Kant's 
exclamation that "modern astronomy has 
annihilated my own importance" ceases to 
have significance when once we know that 
with countless infinitudes encountered in 
time and space our faculties are competent 
to deal, 

'* Times unending 
Comprehending^ 
Space and worlds of worlds transcending." 



68 THE NEW INFINITE AND 

We desire no instauration of the shallow 
and timid humanism that derived its estimate 
of man from a geocentric theory of the uni- 
verse, cried alarm at the crumbling of a 
Mosaic cosmogony and shudders still at the 
shrinking of the earth to a pebble in the 
cosmic perspectives opened to the view by 
modern science. Bigness does not daunt 
Mathesis; she seeks it; vastness is the aether 
that sustains her wing. In her modern doc- 
trine of infinite manifolds, of which I am here 
trying to give a rather slight indeed but hint- 
ful sketch, she has extended the dominion 
of logic far beyond the utmost borders of 
finite things out into the realm of transfinite 
reality. And when, if ever, theology learns 
to follow thither, when if ever, she acquaints 
herself with the procedures of science there 
and learns to contemplate the innumerable 
infinitudes that science can understand, she 
will find that the hierarchy they constitute 
is a ladder for her, an endless ladder by 
which she may ascend higher and higher into 
a better and better sense of what she ought 
to mean by her own Infinitude, which at once 
o'ertops and includes them all. 

At this point the reader may naturally 



THE OLD THEOLOGY 69 

desire to enter the discussion and have a 
share in shaping its course. I have, he may 
wish to say, now acquired a pretty clear con- 
ception of what science means by an infinite 
manifold; I have grasped the abstract idea 
and have seen it realized and illustrated in 
a variety of concrete examples; I am now 
prepared to find other examples for myself, 
for I am beginning to see that such multi- 
plicities compose the intelligible portion of 
the embracing world, that they are literally 
omnipresent, that even in the surface of 
common life and common thought they gleam 
here, there and yonder like shining bassets 
of gold. But I do not see, he may say, that 
they are not of a single type; I have not 
glimpsed the ladder; I am far from seeing 
that, in respect of dignity, they dispose 
themselves rank above rank in a hierarchy 
without a rank supreme. 

These words of the reader imply a legiti- 
mate demand, which must now be met — met, 
that is, in so far as circumstances will allow, 
for the matter is pretty subtle, involving 
some technical considerations known only to 
mathematicians, and does not admit of pub- 
lic presentation in a line. It is possible, 



70 THE NEW INFINITE AND 

however, without too many words, so to 
delineate the matter as to give what is suffi- 
cient — a realizing sense of its truth. For 
this purpose I must ask the reader to look 
again at the infinite manifold denoted above 
by (W). It is a homely affair. How dreary 
it looks and commonplace. But let us not 
be disheartened. We are going to see that 
it has beautiful aspects not yet disclosed, a 
dignity and character to quicken the pulse 
of our thought and win our admiration; for 
we are going to see that it belongs to an 
immense family of similar manifolds, many 
of them of singular beauty, — a countless host 
of them — , and that this homely one has the 
distinction and honor to represent the type. 
Of this family many members differ so much 
in appearance as to have concealed for thou- 
sands of years the deep similitude that makes 
them kin. Why is it, I wonder, that things 
are not what they seem? Is it because, if 
they were, life would be void of its finest 
interest — the zest of research, the joy of dis- 
covery, the surprise and delight of detecting 
the hid? Perhaps so, but let the question 
pass, and attend very sharply to what is now 
to be said. Any ensemble or manifold of 



THE OLD THEOLOGY 71 

elements such that a one-to-one correspond- 
ence can be set up between them and the 
numbers composing the manifold ( W) is said 
to belong to the type of infinite manifolds 
represented by ( W) . This type has a beau- 
tiful designation — it is called the Denumer- 
able Type : the manifolds belonging to it are 
called denumerable infinities or denumerably 
infinite manifolds or classes. Their name is 
legion. One of them, as we have seen, is the 
manifold of even integers, above denoted by 
(P); obviously another is the ensemble of 
odd integers ; another may be got by taking 
from ( W) for elements, say the number one, 
then a million, then a million million, and so 
on; it being thus evident that (W) contains 
countless denumerably infinite parts. But 
let us go outside of (TF). Consider any 
straight line L running endlessly up and 
down — zenith-ward and nadir-ward — pierc- 
ing or passing the stars, like a thread 
stretched through the universe of space. 
Consider the ensemble of miles it contains. 
On it choose a starting point 0. Conceive 
as marked by 1 the end of the first upward 
mile, by 2 the end of the first downward mile, 
by 3 the end of the next upward mile, by 4 



72 THE NEW INFINITE AND 

the end of the next downward mile, and so 
on and on. Will the integers fail? No, you 
say. Will thread length fail? Again you 
say no. So, then, the law of correlation 
holds, the required correspondence is estab- 
lished between the mile posts of L and the 
integers of (WT). Obviously the same would 
be the case if we chose any other unit of 
length. Accordingly, we see that the en- 
semble of unit lengths composing a thread 
or course that traverses endlessly the 
abysses of Space is an infinite manifold of 
the denumerable type. To the same type 
of infinitude plainly belongs the aggregate 
of minutes or hours or centuries in the 
stretch or course of Time conceived as run- 
ning eternally backward and eternally for- 
ward. Well might the apostle exclaim that 
one day is with the Lord as a thousand years, 
and a thousand years as a day. No wonder 
Lucretius could say that, however many 
years you may prolong your life, you can 
not diminish by a single jot the length of 
time you will be dead. Without knowing it, 
these men were thinking in terms of denu- 
merable infinitudes. They did not indeed 
understand the matter scientifically, but they 



THE OLD THEOLOGY 73 

felt it, and in their utterance is the throb 
of its mighty power. I wish now to present 
what is — if one will but ponder it till it is 
clearly seen in the light of meditation — 
perhaps the most impressive, certainly the 
most astonishing, known example of the type 
of infinity here in question. Consider the 
totality of rational fractions, of those frac- 
tions, that is, whose terms (numerators and 
denominators) are integers, or whole num- 
bers. Take any two integers, say 3 and 4, 
and reflect a little upon the multitude of 
fractions that lie between, being greater, 
that is, than 3 and less than 4. Take any 
two of these; between them there is another 
and another; between these, another and 
another; and so on forever. How thickly 
they are crowded together ! More numerous 
than the sands of the sea, than the drops in 
the ocean, for these sands or drops, if 
arranged in a row, would not go on forever. 
In the interval between any two consecutive 
integers stands, then, a countless crowd of 
fractions. Do but reflect and reflect again 
upon the amazing multitude: an infinite host 
in each interval of an infinite host of inter- 
vals. Surely we have here — have we not.? — 



74 THE NEW INFINITE AND 

in this infinity of infinities of rational frac- 
tions an overmatch for the mere ensemble 
(W) of integers. Undoubtedly it seems so. 
But it is seeming only: the appearance 
deceives. The imposing array of all the 
fractions, as soon we shall see, belongs to 
the denumerable type of infinitude. Nay, we 
may even throw all the integers and all the 
rational fractions together, and then show 
that the new multitude is, in respect of 
multiplicity, perfectly matched by the array 
of integers alone, notwithstanding these seem 
in comparison so few and scarce. Let us 
prove this astounding fact, for it is but a 
sample — a model, if you please, or pattern — 
of surprising relationships literally saturat- 
ing the subject-matter of theology and there 
awaiting disclosure for the enlightenment 
and edification of man. The argument is 
easy to follow. Take a fraction at random, 
say % ; note that the sum of its terms — its 
term-sum — is an integer, in this case 5 ; note 
that there are other fractions having the 
same term-sum; arranged in the order of 
increasing numerators, they are: Y^^ %, 
%, Yi. Any other integer will similarly give 



THE OLD THEOLOGY 75 

rise to a set of fractions, which we may 
arrange in similar order. Thus the term- 
sum 2 gives (a) : %. The term-sum 3 gives 
(b) : Yz^ %. The term-sum 4 gives {c) : 
Vs^ %5 %. The term-sum 5 yields (d) : 
■J^? 73? /^j Yi' The term-sum 6 furnishes 
(e): Voj %, %, %, %. And so on forever. 
Observe that this procedure is one that 
sooner or later presents us with any fraction 
whatever that we may designate. Whole 
numbers appear among the fractions, as 
% or %, for example, and a same integer is 
repeated, as % and %, for example; and a 
same fraction appears repeatedly, as %, %, 
for example. We agree, however, to take 
each but once in the matching process now 
to follow. Bear in mind what we are to show : 
it is that the integers of (TF), taken alone, 
perfectly match, in one-to-one fashion, all 
the rational fractions and all the integers 
taken together. The correlation proceeds as 
follows: pair 1 of (W) with Yi of (a) ; next 
pair 2 and 3 of (W) respectively with % 
and /i of (&) ; next pair 4 and 5 of (W) 
respectively with % and % of (c), omitting 
% as a repetition of /4 already paired; and 



76 THE NEW INFINITE AND 

so on and on. We thus get the following 
scheme of one-to-one association: 

1. H; ^, %. 3, %; 4, %, 5, %; 6, I/4, 7, 2^ g, 3^, 9, %; 



Observe that the law of procedure matches 
each integer of (W) with some definite 
fraction or whole number, and each fraction 
or whole number in the grand totality of 
fractions and whole numbers with a definite 
integer of (W). The result, then, is this: 
that grand totality, embracing, as we saw, 
an infinitude of infinitudes of things, so far 
surpassing, it seemed, in elemental wealth the 
manifold of integers, is nevertheless perfectly 
matched by it in that regard, owns precisely 
the same Mdchtigkeit, and is thus only an 
exceptionally impressive member of the great 
family or type of Denumerable Infinity. 

On discovering results so astonishing as 
the one I have just now presented, it is little 
wonder that mathematical students of the 
subject suspected for a time that possibly 
all thinkable or discoverable infinitudes would 
be found upon examination to be of one 
family, of one and the same type — the denu- 
merable type — 5 however much one infinity 



THE OLD THEOLOGY 77 

might seem to surpass another in wealth of 
elements. But the suspicion was short-lived : 
it was soon discovered that everywhere round 
about us there are innumerable infinitudes of 
higher type — infinitudes, that is, such that 
you can no more exhaust the wealth of one 
of them by removing from it a denumerable 
infinity of its elements than you can exhaust 
a denumerable infinitude by taking from it a 
finite collection however large. Indeed it is 
now well known that the denumerable type 
is the lowest type of infinite manifolds and 
that above it, as I have said, there rise in the 
world of thought an endless scale of types. 
I regret the necessity in this connection of 
having to request the reader, if he be not a 
mathematician, to accept a few mathematical 
facts or propositions on the authority of my 
report, for to prove all of them here would 
both expand this volume beyond desirable 
limits and include in it argumentation of a 
kind too technical for the general reader, 
whatever his abilities or attainments in other 
fields. The reader knows that besides the 
rational numbers, above considered, there 
exist what we call, somewhat unhappily but 
for good historical reasons, irrational num- 



78 THE NEW INFINITE AND 

bers. He knows that these are such as V2, 
the number denoted by tt — ratio of the cir- 
cumference to the diameter of a circle — , the 
number denoted by e — base of the Naperian 
system of logarithms — and many others 
equally familiar. He probably does not 
know — what is nevertheless true — that the 
irrationals are, unlike the rationals, not 
denumerable; they are too numerous for 
that — a fact that mathematicians have 
rigorously demonstrated ; the irrationals 
constitute an infinite manifold of higher rank 
or type than the denumerable type; if from 
the totality of irrationals we take away a 
denumerable infinitude of them there will 
always remain infinitely more irrationals 
than we have taken away. Nay, this will be 
true if we take away, not merely one denu- 
merable infinitude of them, but another such, 
then another, and so on endlessly, thus 
removing a denumerable infinitude of denu- 
merable infinitudes ! The original ensemble 
remains absolutely undiminished — its wealth 
of elements, its Mdchtigkeit, or "power", its 
dignity, its rank or type, is the same as 
before the great removal or decimation. 
This wonderful type of infinitude has, like 



THE OLD THEOLOGY 79 

the other type considered, a fine name: it is 
called the Type of the Continuum; it is so 
called because the totality of points in a 
continuous line of any length, however short, 
is a familiar example of an infinite manifold 
belonging to the type in question. "What 
is it that you say?" may interject a reader. 
"Do you mean to tell me the ensemble of 
points in a little short line or the ensemble 
of instants in a little stretch of time would 
not be exhausted if we could take away from 
it a denumerable infinitude of points in the 
one case or of instants in the other?" The 
answer is, I do: such a taking away would 
not only not exhaust the ensemble but it 
would not even diminish its wealth of ele- 
ments — no, not if the great subtraction were 
repeated a billion times in each second in 
an endless succession of seconds ! Assuming 
that this is mathematically sound, which it 
is, is it unreasonable to say, as I do say, that 
our amiable theological friends and guides, 
in preparing to instruct regarding the theo- 
logical Infinite, which is over and above all 
other infinitudes, would do well to gain some 
insight into the wonders of those that are 
below? Is it unreasonable to contend that 



80 THE NEW INFINITE AND 

a course of lectures in this matter ought to 
be regularly provided in our theological 
seminaries? Personally I have no doubt at 
all that a competent student of theology 
could be, not only much informed, but 
thrilled, joyed, and inspired, by the marvels 
of insight and perspective that such a course 
could open to his gaze. That, however, by 
the way. Members of the great family of 
the Continuum type of infinity are omni- 
present in our world. One of them is the 
manifold of rational and irrational numbers 
taken together; another is the collection of 
instants in a second or a thousand years; 
another is the ensemble of points in a sphere 
or in the universe of space; another is the 
ensemble of angles among the lines of a sheaf 
or the ensemble of the lines themselves; 
another is the pencil of planes having a line 
in common, or the collection of spheres 
centered at a point, or the totality of rela- 
tions between points of time and states of 
the world, or the aggregate of possible 
motions, or the group of possible poses of 
a tiger or a statue, or the multitude of 
simple equations showing how two variables 
may change together, or the multitude of 



THE OLD THEOLOGY 81 

spaces like ours that coexist in a space of 
higher dimensionality; and so on and on 
endlessly and forever. Is the Continuum 
type the next above the Denumerable type? 
Probably so but no one knows. Whoever 
answers the question will thereby immor- 
talize his name. 

Shall we proceed to infinite types that are 
superior to that of the Continuum in respect 
of dignity or rank? It were possible so to 
do but the way is steep. I fear to weary the 
reader by too much demonstration of what 
he is now, I venture to hope, prepared to 
assume. For as we go up from level to level 
in the ever ascending scale of more and more 
embracing infinitudes, the thought becomes 
abstracter and abstracter, the heights 
dizzier and dizzier. At least for the present 
we have perhaps climbed enough. At 
another time, when our lungs have become 
accustomed to the tenuous air, the ascent 
may be resumed. Here and now it is suffi- 
cient to know that the hierarchy exists, that 
each rank includes all ranks below it, and 
that, taken together, these ranks above 
ranks of Infinitude, amenable to the ways of 
Reason, constitute, as we have said, an end- 



82 THE NEW INFINITE AND 

less ladder, an ever rising scale, along which 
the subtle process of Idealization, with the 
velocity of spirit, proceeds upward forever, 
attaining never a summit, for there is no 
summit, but intimating, indicating, and ever 
approximating, an outlying Limit, supernal, 
above all ranks, embracing all, reflecting the 
glory of all — the Infinite of theology. 

The foregoing sketch has, I trust, made it 
fairly clear in a general way that in the 
study of the infinites of science, which are 
infinite ideas, and not elsewhere, there is a 
scientific way to the meaning of theology's 
Infinite, which is, not an infinite idea, but 
something more supreme — an infinite ideal. 
In "a general way," I have said, for the 
considerations adduced have been, in the 
main, pretty broad and general. I wish now 
in approaching the end — for this writing 
must terminate — to descend to particulars 
and to show by some concrete examples 
how the study in question can render the 
service claimed. 

As every one knows, the indictment that 
men of rationalistic temper, including for the 
most part scientific men, have brought 
against theology, is not the same as their 



THE OLD THEOLOGY 83 

objection to religion. It is very far from 
the same. Their indictment of theology 
charges that theology is not coherent, that 
it is replete with internal contradictions, that 
it thus fails to meet the rightful demand of 
intellect for harmony, and so fails to meet 
the standard essential alike to science and to 
art. The indictment is fatal unless the 
alleged contradictions, familiar to all, can 
be purged away or else transcended. Broadly 
speaking they are of two kinds — foreign and 
domestic — contradictions, that is, arising 
from theology's use of assumptions or postu- 
lates that, however available elsewhere, are 
entirely outside theology's proper domain, 
and contradictions that do not arise from 
imported postulates but present themselves 
properly in theology's subject-matter as seen 
from interior but inadequate or fragmentary 
points of view. Contradictions of the for- 
eign variety may, I think, be gradually 
purged away by ridding theology of imported 
postulates ; contradictions of the domestic 
kind may, I am equally confident, be tran- 
scended more and more by seeking view- 
points more and more commanding. 

I wish to indicate what appears to me to 



84 THE NEW INFINITE AND 

be the right manner of dealing with these 
two comprehensive varieties of theological 
contradiction or difficulty. And first a word 
respecting the foreign kind. These are like 
the contradictions that would defeat the 
ends of justice if, in the trial of a case at 
law, it were assumed and held throughout 
that all witnesses are honest or that none 
can be mistaken; or like the hopeless con- 
fusion that would result to the science of 
hydraulics, did the student adhere to the 
postulate, as universally valid, that water 
runs down hill; or like the confusion that 
would arise in chemistry if the chemist 
assumed that the rate of chemical reaction 
depends solely upon the kind, and never upon 
the amount, of the substances involved; or 
like the contradictions that would confound 
the theory of functions if it laid down as a 
postulate that every continuous function 
possesses a derivative; or like the contra- 
dictions that would stay the progress of 
geometry did this science assume that all 
geometric constructions are feasible with 
ruler and compasses ; or, in general, like the 
entanglements that must always ensue when- 
ever, in any field of thought, we consciously 



THE OLD THEOLOGY 86 

or unconsciously employ one or more postu- 
lates that, though valid elsewhere, in a 
more restricted field, are not valid in the 
field of our actual operation. What is the 
remedy? It obviously is to reject the postu- 
lates whence the entanglements arise. The- 
ology is, then, confronted with the task of 
weeding her garden of alien postulates. The 
task is diflScult. Theology must ascertain 
what her postulates are — what assumptions 
she actually makes — and this is not easy to 
do, for assumptions are sly — they do not, 
as a rule, loudly proclaim their arrival or 
their presence. Moreover, when once they 
are ascertained, there remains the difficult 
problem of discrimination: which of them 
are legitimate, or domestic, and which are 
illegitimate, or foreign? 

Perhaps the most noxious, certainly the 
most flagrant, of theology's foreign postu- 
lates — one that has engendered endless con- 
fusion within and brought from without no 
end of ridicule — is the hoary assumption 
that, in ever}^ subject-matter or field of 
thought, the whole is greater than the part. 
It is not perhaps strange that this so-called 
axiom became an article of universal belief 



86 THE NEW INFINITE AND 

in the early stages of human development, 
when the interests of men were of necessity 
confined to the concrete things of sense, but 
it is strange, very strange, that the belief 
persisted as universal throughout the his- 
tory of thought despite the fact that the 
subject-matter of thought is everywhere and 
continuously vocal with its denial. Except 
in the case of mathematicians and some 
philosophers, the proposition is even today 
universally held to be universally valid. The 
fact is, however, as we have abundantly seen, 
that the proposition, instead of being univer- 
sally true, is generally false. The discovery 
of this fact — the discovery, about fifty years 
ago, that, instead of being an essential prin- 
ciple of reason, the proposition merely serves 
as a principle of classification, as a logical 
blade, we may say, sundering the universe 
of thinkable things into two components; 
the discovery that one of these — the world 
of finite things — is composed of wholes to 
which the proposition does indeed apply 
without exception, but that the other com- 
ponent — the world of infinites — is composed 
of wholes for which, without exception, the 
proposition is false; thie discovery that the 



THE OLD THEOLOGY 87 

latter world, the world of infinite wholes, is 
par excellence the domain of reason, and 
that, in respect of content, it is immeasur- 
ably richer than the world of finite wholes : 
that discovery I judge to be second in 
importance, for the future of thought, to no 
event in the history of mankind. And 
auspicious for theology will be the day when 
she really discovers that Discovery, when she 
really learns that her subject-matter belongs 
strictly to the world of infinite wholes, and 
accordingly relinquishes, as then she will, the 
ancient dogma of whole and part as alien to 
her field. 

Let me give an example to illustrate the 
great emancipation that will ensue. Not 
long ago in a western city of the United 
States a great orator, speaking of the 
dogma that the persons of the Trinity are 
each Almighty and yet together constitute 
but one Almighty, speaking of the doctrine 
that each of the Persons is equal to the One 
composed by all of them, evoked general 
applause from a vast audience by character- 
izing the venerated creed as "infinitely 
absurd." Why.^ Because the speaker and 
his hearers tacitly assumed that as a matter 



88 THE NEW INFINITE AND 

of course the whole must exceed the part. 
And why does not theology explain the 
difficulty? Why does she content herself 
with avowing that the alleged composition 
of the Trinity is an "incomprehensible 
mystery"? Because she, too, makes the 
same assumption. And yet it is not the 
dogma but the orator's characterization of 
it that is "infinitely absurd." Let us see 
clearly that this is so. It is plain that we 
have here to do with the structure of infinite 
manifolds. More than fifty years ago, that 
profound mathematician, philosopher and 
theologian, Bernhardt Bolzano, pointed out 
that "there are points of view from which 
we perceive in God an infinite multiplicity 
{unendliche Vielheit)^ and there are no 
other viewpoints from which we attribute 
infinity to him." "Ich sage nun," he adds 
in explanation, "wir nennen Gott unendlich, 
weil wir ihm Krafte von mehr als einer Art 
Zugestehen miissen, die eirie unendliche 
Grosse besitzen. So miissen wir ihm eine 
Erkenntnisskraft beilegen, die wahre AU- 
wissenschaft ist, also unendliche Menge von 
Wahrheiten, weil alle ueberhaupt, umfasst, 
und so weiter." The key-word, as the con- 



THE OLD THEOLOGY 89 

text shows, is the term unendliche Menge^ 
infinite ensemble or multitude or manifold. 
Now consider, for example, the following 
infinite manifolds : the totality E of even 
integers, the totality O of odd ones, the 
totality F of fractions having integers for 
their terms. Denote by M the totality of 
rational numbers. M, you see, is composed 
of the elements of E and and F. M con- 
tains each of these elements once and only 
once and contains nothing else. Any child 
knows that there is an even integer for every 
odd one, an odd integer for every even one, 
and so, it is plain, E and are equally rich 
in constituents. Recall, as we proved some 
pages back, that the same is true of E, which 
we there denoted by (P), and our old friend, 
the ensemble {W), Hence it is true of O 
and (TF). It is true also of F and (TF) for 
we saw that we could match the integers with 
the rational fractions. Hence it is true of 
E and O and F and ( W) : these are equally 
rich in elements. But did we not show the 
like for M and ( W) ? We did, whence it 
follows that, in wealth of constituents, E and 
O and F and M are exactly on a par: they 
belong to the same type of infinitude. It 



90 THE NEW INFINITE AND 

happens that it is the denumerable type but 
that fact is not important. What is impor- 
tant is now obvious : it is that we have here 
three infinite manifolds, JE, O, F, no two of 
which have so much as a single element in 
common, and yet the three together consti- 
tute one manifold M exactly equal in wealth 
of elements to each of its infinite components. 
Have we proved that there is a Trinity 
composed of three components related to one 
another and to the Trinity as the dogma 
asserts? No. We have proved that the 
conception of such a Trinity, instead of 
being rendered absurd by a so-called axiom 
having no application to infinite manifolds, 
is rigorously thinkable, perfectly possible 
and rational, and that our brilliant orator 
was indeed in this instance an ass. Far from 
being absurd, the conception would be rigor- 
ously thinkable — as mathematicians know 
and as the reader of these pages ought now 
to see — even if the One it contemplates were 
asserted to be, instead of a trinity of per- 
sons, a multiplicity of order four or a mil- 
lion. Nay, an infinite / of even the lowest 
type always contains, not merely two or three 
or a million components each equal to it in 



THE OLD THEOLOGY 91 

plenitude of elements, but an infinity of such 
components. The like is equally true of the 
infinites of whatever type in the endless scale 
of types. Must we suppose the truth to fail 
in the case of theology's Infinite, the great 
ideal towards which the others mount for- 
ever, ever rising from the level of one 
sublimity to another yet more sublime.'^ Is 
the nature of an ideal inferior to that of the 
ideas it hovers above .^ Is perfection inferior 
to approximation? 

For another example of the great emanci- 
pation that will come to theology the moment 
she casts off the yoke of the Vhole-part 
axiom' that has hitherto hampered the 
proper movement of her thought, witness 
the possibility of handling anew and radi- 
cally the question of Omniscience in relation 
to that of Freedom. I purpose to treat here 
briefly a single phase of the matter, a cen- 
tral difficulty of it, familiar to every one. 
If, so the dialectic runs, God is omniscient, 
he knows what I shall do, he knows my 
future, he knows, before I make decisions, 
what they will be, and if he knows that, then 
to trust the feeling that I am free to choose 
is "to cheat the eye with blear illusion." On 



92 THE NEW INFINITE AND 

the other hand, if God does not know all 
future events, he is not omniscient and the 
supreme dignity ascribed to him is thereby 
impaired. The argument is specious but, 
as we are going to see, it is false. The 
problem is to reconcile, not Freedom and 
Omniscience, but Freedom and the Dignity 
of omniscience. Let it be granted that, if 
you are free, God is not omniscient. It does 
not follow that he is less in respect of 
dignity than if he were omniscient. Sup- 
pose two Beings : one of them capable at once 
of knowing all and of not knowing all; the 
other one capable of knowing all but inca- 
pable of not knowing all. Are they coequal 
in respect of dignity .^^ No, you will probably 
say, the latter one is, in respect of dignity, 
distinctly inferior to the former. If that be 
your answer, I shall agree. I do not, how- 
ever, intend to depend here upon such 
intuitive estimates of worth. I purpose to 
prove that a Being of infinite knowledge may 
have all the Dignity of Omniscience without 
being omniscient. To do so, we must again 
have recourse to the nature of infinite mani- 
folds. Instead, however, of employing, as 
I might, any of those hitherto presented, I 



THE OLD THEOLOGY 93 

shall ask you to consider a more shining one, 
one that appeals to the imagination like the 
open sky. 

Let n be an entire plane ; it bisects the 
universe of Space. I must ask the reader 
to assume — for it is true and might easily 
be shown did space allow — that a one-to-one 
correspondence, of the kind with which he is 
now familiar, can be established between the 
totality T of points in space, those of 77 
included, and the totality S of points on 
either side of 77. Note carefully that, as 
77 is any plane, the correspondence will be 
equally possible, if 77 be moved parallel to 
itself any finite distance. Now suppose each 
point of the infinite totality T to represent 
an element e of knowable reality, and denote 
by d the element of spiritual dignity that 
attaches to knowledge of e. At once we see 
that a knowledge Ks extending to all and only 
the elements e of the part-totoliiy Se of 
knowable reality represented by the points 
of S is precisely as rich in elements d of 
scientific or spiritual dignity as is a knowl- 
edge Kt extending to all the elements e of the 
whole-totalitj Te of knowable reality repre- 
sented by the points of T, Now suppose that 



94 THE NEW INFINITE AND 

Te is the whole of knowable reality, then Kt 
is omniscience. We behold the astounding 
fact that omniscience does not by even the 
smallest mite surpass in dignity the partial 
knowledge Ks. But how, one may ask, does 
this fact advance the solution of our prob- 
lem? How does it enable us to maintain the 
doctrine of Freedom and still attribute to 
God a dignity of knowledge equal to the 
Dignity of omniscience? For, our inter- 
locutor will say, knowledge is related to 
Time, it is of things that have been or are 
or will be; omniscience must cover them all, 
it must extend at once through Past, Present 
and Future ; whilst Freedom means that you 
and I are capable of determining what the 
Past is to be by choosing in the Present, for 
actualization, from among the possibilities 
that constantly descend upon us out of Time- 
to-come like in-rolling waves of an infinite 
sea. But, our critic will urge, such capa- 
bility does not exist if omniscience cover the 
Future and if, accordingly, the destiny of 
possibilities is determined before they arrive. 
And what, he will say, is to be said of the 
Dignity of knowledge that, though covering 
the Past, does not extend to all events that 



THE OLD THEOLOGY 96 

are yet to be? In answer let me ask the 
reader to change a little the imagery em- 
ployed in our previous argument: let us 
suppose that 77 is, not as before an ordinary 
plane bisecting Space, but what we may call 
a moving Time-plane — the Present — bound- 
ing off the Future from the Past. Behind 77 
is an eternity of time that has been; before 
it, an eternity of time that will be. The two 
eternities, regarded as manifolds of the 
things they contain, are infinitudes of the 
same type, and — what is important to note — 
they are, as infinites, each of the same type 
as the one Eternity that together they con- 
stitute. In respect, then, of Dignity of 
knowledge, complete knowledge of the eternal 
Past is not inferior to knowledge extending 
both backward and forward, covering the 
composite Eternity of both Future and Past. 
It is important to observe that the proposi- 
tion continues to be true as the Time-plane 
n — advancing forefront of Universal His- 
tory — with infinite range and sweep of wing 
moves continuously forward ; for, though the 
Past, as we say, thus grows continuously 
longer and longer, and the Future shorter 
and shorter, yet the two eternities keep 



96 THE NEW INFINITE AND 

forever their common membership in the type 
of infinity to which they belong. And so it 
appears that Freedom is entirely compatible 
with the Dignity of omniscience, though it 
is not compatible with Omniscience itself. 
I fancy that many a spiritual-minded de- 
fender of the doctrine of Freedom would 
find it no great hardship to give up that of 
Omniscience, seeing that the sacrifice does 
not involve denying to God the Dignity of 
omniscience. Such a defender could say: 
'I maintain that, to the Supreme Intelligence, 
the Past alone is completely known; I main- 
tain that the Future is not completely 
known; I maintain that, as the Present 
moves on continuously forward into the 
realm of potentialities, the eligible gets 
sifted, becoming in part the chosen, that part 
of the possible and unknown becomes the 
actual and known; I maintain that mean- 
while the infinite Dignity attaching to 
knowledge of the growing Past remains 
forever invariant, equal absolutely to the 
dignity of omniscience itself; and that Free- 
dom remains.' Many will be glad to know 
that such a dogma, whether true or not, is 
at all events, thanks to the nature of infinite 



THE OLD THEOLOGY 97 

manifolds, free from internal contradiction 
and may, therefore, be held without surren- 
dering reason. Unless I am much mistaken, 
the distinction, herewith mathematically 
drawn, between the Dignity of omniscience 
and Omniscience itself, whereby we may 
affirm the doctrine of Freedom without im- 
puting to God's knowledge a Dignity less 
than that of knowing all, is fundamental. I 
leave it to the reader to see, in the light of 
his own reflection, that a similar distinction 
is available, if required, in dealing with other 
attributes — Omnipotence, for example, or 
Omnipresence — commonly ascribed to Deity. 
I purpose to deal here with Omnipresence 
but from another point of view. 

Our task is to vindicate the logical possi- 
bility of Omnipresence — not by such inade- 
quate analogies as immortal Bruno, for 
example, ingeniously employed in comparing 
it to a voice audible at every point of a 
room — ^but by considerations bringing it 
strictly within the category of doctrines 
rigorously thinkable. Consider a sphere. 
Let it be so small that, even if it were a 
brilliantly colored globe, the most powerful 
microscope could not reveal its presence. 



98 THE NEW INFINITE AND 

It is to be carefully noted that the following 
statements regarding it are absolutely inde- 
pendent of its size, and remain true if it be 
supposed shrunken to any degree of parvi- 
tude, however small, so long as it has not 
vanished utterly. Denote by s the totality 
of points within the tiny sphere, and by S 
the ensemble of all the other points of the 
whole of Space. In the course of recent 
years and by means within the grasp of the 
average student a little disciplined in the 
ways of rigorous thought, it has been demon- 
strated that there are precisely as many 
points in s, as in S, and that the former are 
joined to the latter in one-to-one fashion 
by relational rays of correspondence. As 
such correlation subsists in countless modes, 
suppose one of them chosen. This done, to 
any point of S, say the center of the sun, 
corresponds a definite point of s; to any 
other point of S, say the center of the moon 
or the mass-center of the Milky Way, corre- 
sponds another definite point of s; and so on 
and on throughout the range of both totali- 
ties : no element of either manifold but it has 
a match or mate in the other and no two 
of either manifold have a common mate. 



THE OLD THEOLOGY 99 

Let no one fail to see clearly that in that 
tiny sphere, too small, mind you, for even 
microscopic vision, small indeed at will, 
there nevertheless exist point configurations 
matching perfectly in detail and every re- 
spect of inner constitution each and all of 
the infinitely inlSnite hosts of point configu- 
rations, minute and vast, simple and com- 
plex, here, there, and yonder, everywhere 
throughout the height and depth and length 
and breadth of Space. We have now only 
to reflect that the same scheme of repre- 
sentation obtains universally, being valid at 
once for all infinitesimal spheres, and the 
truth dawns that the Whole really is incar- 
nate in every Part — the Emersonian apho- 
rism that "the universe contrives to integrate 
itself in every smallest particle" being thus 
completely justified on scientific ground. 
But this is yet not all. The universe is 
dynamic, charged throughout with innumer- 
able modes of motion. Each point, however, 
of any moving thing — an ion of gas, a 
vibrating fiber of brain — is represented by 
a corresponding point in s, and so within 
the tiny sphere — indeed in every room how- 
ever small — the whole dynamics of the uni- 



100 THE NEW INFINITE AND 

verse is depicted completely and coenacted 
by motion of points and transformation of 
point configurations. There in miniature 
proceed at once the countless play and inter- 
play of every kind of motion, small and 
large, simple and complex, the quivering 
dance of the molecule, the wave and swing 
of universal aether. 

" Wie AUes sich zum Ganzen webt ! 
Eins in dem andern wirkt und lebt! 
Wie Himmelskrafte auf und nieder steigen 
Und sich die goldnen Eimer reichen ! 
Mit segenduftenden Schwingen 
Vom Himmel durch die Erde dringen, 
Harmonisch air das All durchdringen !" 

The immense labor to be performed by 
theology in eradicating from the proper 
domain of her study the whole-part dogma 
with its ubiquitous progeny of confusion ; and 
the light, the freedom, and the power that 
will more and more accrue to her as the 
work proceeds : these are not the end but 
are only the beginning of her emancipation. 
For the whole-part "axiom" is not the sole 
postulate of the imported kind that troubles 
her thought. Once she seriously enters upon 
the search, she will find that there are others. 
I have already repeatedly pointed out that 



THE OLD THEOLOGY 101 

the subject-matter of her thought — ^the 
realm of transfinite reality — presents infini- 
tudes in a hierarchy without a summit. As 
she passes upward in her study from level 
to level, she will find that a postulate avail- 
able at a given elevation may have to be 
relinquished on passing to a higher rank. 
For example, nothing can seem more natural 
or axiomatic than to suppose that, if we 
have any manifold of elements, these are 
capable of being arranged in a row, like 
marbles, so that after each there is a next — 
none, that is, between. Nevertheless, as 
mathematicians have recently ascertained, 
that seemingly universal possibility is re- 
stricted very narrowly. The possibility — 
let us call it the postulate of Nextness — 
does indeed hold for all infinite manifolds of 
the Denumerable type but it fails utterly 
for every manifold of the Continuum type 
or of any higher type. The employment of 
foreign postulates is equally disastrous 
whether the importation be, as in case of the 
whole-part postulate, from the realm of the 
finite, where it is valid, to the realm of the 
infinite, where it is not, or, as in case of the 
nextness postulate, from a type of infinitude, 



102 THE NEW INFINITE AND 

in which it applies, to a higher type, in which 
it does not. It is now, I believe, sufficiently 
evident that eternal vigilance against the 
admission of alien assumptions is part of 
the price theology must pay for freedom, 
for freedom, that is, from the fatal presence 
of internal confusion. 

Before undertaking to deal with the other 
variety of contradictions — the kind, I mean, 
that arise properly, because they arise from 
domestic or native postulates — , I desire to 
allude briefly to another mathematical idea, 
one that is destined, I believe, as the eye 
becomes more and more adjusted to its light, 
to be of great service in theology, especially 
enlarging her conception of the Conceivable, 
and serving to bring not only the attribute 
of Omnipresence, with which we are here 
further concerned, but kindred attributes as 
well, strictly within the category of intelli- 
gible ideals. I refer to the radiant concept 
of Hyperspace. Only a generation ago this 
concept was regarded even by mathemati- 
cians — most adventurous of men — as vision- 
ary and vain. Meanwhile it has advanced 
so rapidly to commanding position that 
today its instrumental value is — strange to 



THE OLD THEOLOGY 103 

say — recognized even in "natural" science, 
by Van't Hoff, for example, in chemistry, 
and by leading physicists in the kinetic 
theory of gases. The statement made by 
Poincare seven years ago before the Inter- 
national Congress of Mathematicians at 
Rome is well within conservative limits : 
"Nous sommes aujourd'hui tellement famil- 
iarises avec cetti notion que nous pouvons 
en parler, meme dans un cours d'universite, 
sans provoquer trop d'etonnement." The 
fact is that the doctrine of hyperspaces 
already exists in a copious and rapidly 
growing literature, flourishes in every scien- 
tific language of the world, and in its essen- 
tial principles has become for mathematicians 
as orthodox as the multiplication table. 
Indeed, as Professor Klein has shown, the 
modern physical theory of Relativity is, in 
point of structure and form, a species of 
four-dimensional geometry. My aim here is 
to indicate how the hyperspace concept 
enables us to show the c one eiv ability of an 
infinite Being being everywhere present in 
an infinite region without being contained in 
it. Anyone who will devote a little time to 
reflecting upon the infinite wealth of points 



104 THE NEW INFINITE AND 

in, say, a straight line L and upon the infinite 
wealth of detectible combinations and inter- 
relations subsisting among them, will dis- 
cover to his astonishment that a linear being 
or intelligence X inhabiting L and in its expe- 
rience strictly confined thereto would have, 
in its own habitation, all the material 
necessary for constructing mathematical doc- 
trines matching completely, in diversity 
and in complexity, all branches of geometry 
and analysis constructible by man, despite 
the immensely superior resources the latter 
seems to have in inhabiting the three-way 
spread of Space. Marvelous as it seems, the 
parity exists. Such a being A, dwelling in 
the midst of such magnificence of subject- 
matter, order, and law, naturally might 
attempt to construct a rational theology. 
If so, it would encounter, among other diffi- 
culties, that of understanding how the 
supreme being it hypothetized could be at 
one and the same time present everywhere in 
the line-world L. Note that, by hypothesis, 
X could have no sense-perception or geometric 
intuition or image of the fact that the infinite 
region or line-world L, in which it lives, 
moves and has its being, is, as we humans 



THE OLD THEOLOGY 105 

happen to know, itself contained or immersed 
in another infinite region of higher order, 
namely, a plane 77; hence X could not per- 
ceive, though it might feel, and it might in- 
deed conceiye^ the fact that the infinite, 77, is 
actually omnipresent to L, every part of this 
line-world being, as we know, completely 
immersed in 77; and so A could not perceive, 
yet after some centuries of theologizing it 
might coTiceive, how the same attribute — 
omnipresence in the line-world L — could 
belong to a being whose reality, whatever its 
nature in other respects, was at least co- 
extensive with the higher world 77. Who 
can fail to see that precisely like reflections 
would be equally pertinent, if we replaced 
the line-world L by the plane-world 77 and 
the latter by Space itself? We live in 
Space — a three-way spread — and encounter 
precisely the same difficulties encountered by 
our linear friend A, and they are surmount- 
able in the same way, namely, by the concept 
of Hyperspace. For this world-creating 
idea, at once exquisite and vast, presents us 
in the first place with a four-way spread, a 
four-dimensional space, *S'4, completely im- 
mersing our ordinary space, being in contact 



106 THE NEW INFINITE AND 

with all its points and present at all of them, 
just as our ordinary space is omnipresent 
to all the elements of the plane-world 77, and 
this, in turn, to all those of L; next, similarly 
related to 1^4, comes a yet higher world S^ ; 
then follow, in order of ascending dimension- 
ality, Sq^ Stj . . ., aSii, . . . and so on end- 
lessly: affording thus conceptual provision 
for the presence everywhere in our dwelling- 
place of a Being whose reality, if you please, 
not only pervades but infinitely transcends 
any assignable space, however high its rank 
in the summitless scale of hyperspatial 
grandeur. Is it a small service to show that 
theology's supreme ideals conform to pat- 
terns woven of scientific ideas? Is it a little 
thing to demonstrate the reasonableness of 
reason's dreams? 

Finally, I come now to the keeping of my 
promise regarding theological difficulties of 
the domestic kind. These are not due to the 
lurking presence of alien postulates, and are 
not to be overcome by the process of casting 
out. They are due to the peculiarly vast 
and complicate character of theology's 
subject-matter, to the great diversity of 
aspects presented by it, and the consequent 



THE OLD THEOLOGY 107 

necessity of examining or beholding them 
from a corresponding variety of partial or 
fragmentary points of view. Such native 
difficulties are to be conquered, progressively 
of course, not by elimination, but by the 
method of surmounting, by the process of 
transcending. What does this method con- 
sist in.f^ What does such transcending mean.'^ 
It does not mean, as commonly supposed, the 
finding of a point of view from which the 
difference of two aspects of a matter shall, 
as this is seen from other points of view, 
seem to disappear, for that would be, not to 
clarify, but to obscure, to disguise fact, to 
hide truth. Transcending does not mean 
that. It means — and the answer is very 
important — recognition of the fact that two 
differing aspects of a matter are indeed, not 
one, but two^ and that the matter is, in truth, 
such as to present them both. It thus means 
submission of the understanding to facts, 
not facts to the understanding, and, in dis- 
course, to speak of a matter as it is and not 
as we may wish it to be. Doubtless the aim 
of science is art but the beauty it seeks does 
not lie in the way of disguisings or mutila- 
tions, for it is the beauty of truth. 



108 THE NEW INFINITE AND 

Before presenting concrete illustrations 
may I outline the matter briefly in abstract? 
Denote by B some being, some complex and 
multi-phased entity, the subject or object of 
thought. In view of some aspect of B we 
construct a theory Ti, which, as we are not 
aware of other aspects, we call a theory, not 
of a phase of B, but of B itself. Some other 
aspect of B, seen at another time by us or 
at the same time by some one else, gives rise 
to another theory T25 which, like Ti and 
owing to the same circumstance, claims to be 
a theory of B; and so on, for other phases of 
B. Let us suppose that the theories have 
been soundly made after the manner of 
autonomous doctrines. Ti, then, consists 
of a definite basal system of compatible 
postulates together with a superstructure of 
rigorously deduced implications. Of T29 we 
must say the same. Each of the theories is, 
accordingly, thoroughly coherent, absolutely 
void of inconsistency among its component 
elements. They do not, however, coincide: 
though having perhaps many propositions 
in common, yet either T contains at least 
one proposition that contradicts some propo- 
sition of the other. Let us suppose, more- 



THE OLD THEOLOGY 109 

over, that each theory is true to the aspect 
that gave it birth: that is, seen from one 
point of view, B appears exactly as Ti 
describes it; from another, exactly as T2 
describes it ; and so on, of course, if there be 
other theories. What happens? This: 
sooner or later, in one or another of the ways 
familiar to students, Ti and T2 get com- 
pared; it is noted that each of them claims 
to be a true theory or account of B; it is 
observed also that in one or more respects 
they are mutually contradictory. What 
follows? It follows that the claim must be 
disallowed in the case of at least one of them : 
regarded as accounts of one object or sub- 
ject, two discordant doctrines may be both 
of them false but they can not both be true. 
But we have seen that each of them is true 
to the B-aspect that gave it birth; yet they 
contradict one another. What is to be done? 
Reject them both? No. The remedy is: 
Keep both and transcend them. Keep them 
both, for the contradiction arises from sup- 
posing them to be speaking of B as a whole, 
which they are really not; it disappears 
when we suppose them to be speaking 
respectively of different phases of B, which 



no THE NEW INFINITE AND 

they really are. The act or process of sur- 
mounting consists — not in constructing one 
theory to cover at once both the aspect 
covered by Ti and that covered by 7^2, for 
that is impossible — the nearest possible ap- 
proach to it would be to construct a theory 
covering the common part (if any) of the 
two aspects, and plainly such a theory would 
consist of the common part, or intersection, 
of Ti and T2: no, the surmounting or tran- 
scending of Ti and T2 consists in recognizing 
once for all that the object B does in fact 
present the two aspects in question and 
thereby validates at once both of the theories 
in question. Do you ask what is thus gained.'* 
I answer that the proposition stating the 
recognition is new ; it is not in Ti nor in T2 ; 
it is not a fact about either of the phases 
dealt with by Ti and 7^2; we have mounted 
higher — the new truth is a truth about B 
itself. 

Is the matter so clear in the abstract as 
not to be impressive? I sincerely hope that 
it is, for in that case it will not require many 
concrete illustrations and of these, moreover, 
the simplest will suffice. I will begin with one 
so simple as to seem trivial. Yet its illustra- 



THE OLD THEOLOGY 111 

tive value is, I believe, very considerable, 
unless our familiarity with the phenomena 
involved, blinds us to their worth. On my 
table lies a slender rod. As seen there, it 
appears to be straight. I place it at a slant 
in a vessel of water. As seen there, it ap- 
pears to be bent. Is the rod straight or 
bent? That is not the question. If it were, 
we should have to invoke the testimony of at 
least another sense, which, however, for the 
purpose of the illustration, I exclude. I am 
admitting vision only. To vision, then, the 
rod presents two contradictory aspects — 
now straight, now bent. Are they, as 
aspects, false. ^ Is either, as an aspect, false .^ 
Neither, as an aspect, is false: as aspects, 
both are true, both are genuine, both actual. 
How surmount them.'^ The answer is by 
recognizing that the rod is such a thing in 
our world that it does, in truth, present to 
vision both aspects — and that recognition is 
a valuable event because it tells a truth about 
the rod, about our world, and about our 
vision. 

For another lean but helpful illustration, 
consider the quadratic expression, x^ — 4. 
If you write, £C^ — 4 = 0, then I can affirm 



112 THE NEW INFINITE AND 

that iT = 2 or that x = — 2 but you are 
right in not allowing me to say that x = both 
2 and — 2 at once. Everyone knows, how- 
ever, how to transcend the seeming necessity 
of the alternation: ^ = 2 or ^ = — 2. We 
do it, that is, by saying that the given 
equation is a thing of which 2 and — 2 are, 
at the same time, roots. Is such surmounting 
merely a trick? On the contrary it is a 
legitimate procedure of thought: the taking 
of both of two things when either is allowed, 
or taking all of many when any is allowed: 
it is the familiar bound of the spirit from 
alternation to conjunction or more often 
from the level of partial dissonance to the 
bridge of an overarching harmony. 

A much more impressive example of such 
surmounting is found in the manner in which 
geometricians deal with the infinitely distant 
region of space. There are, as the reader 
may know, various kinds of geometry of 
space. In one of these the infinite region 
of space presents one aspect; in a second, 
a second aspect; in a third, a third; and so 
on indefinitely. These various aspects differ 
among themselves immensely — they are even 
inconsistent with one another or mutually 



THE OLD THEOLOGY 113 

contradictory and exclusive. Thus, in what 
is called projective geometry, alluded to 
previously herein, the aspect presented by 
the infinite region of space is that of a plane 
all of whose points are infinitely far away; 
in what is called inversion geometry, which 
I need not here explain, the same infinite 
region appears to be a point where, curiously 
enough, all lines of space seem to meet and 
pass. What do geometricians do in the 
matter of such conflicts, at first so shocking.'^ 
Do they reject the aspects as false because 
they are mutually incompatible.? Far from 
rejecting any of them, they keep them all, 
use them all, rejoice in them all, and — tran- 
scend them all. But how transcend.? Again 
the answer is — and how replete with sig- 
nificance for theology! — the answer is that 
geometricians simply recognize that the 
infinitely distant portion of space is, whether 
one likes it or not (and geometricians do like 
it), in its own nature just such a thing as 
to present in fact all the diverse aspects in 
question, and so to validate — at once, mind 
you — all the geometries in question. 

Similar matter presses from every side; 
but enough has been said, I trust, to indicate 



114 THE NEW INFINITE AND 

the method that mathematics would recom- 
mend for dealing with theological difficulties 
of the domestic or native kind. As theology 
proceeds with her great enterprise of advanc- 
ing the science of Idealization, as in particu- 
lar she continues to clarify and estimate the 
significance of the supernal ideals that she 
ascribes as attributes to Deity, she is des- 
tined to discover that those attributes, how- 
ever indubitable or undeniable they may be 
when regarded singly, yet, taken together, 
involve essential and ineradicable incompati- 
bilities of thought, and, therefore, must 
finally defeat every possible effort to com- 
bine them in one self-consistent body of doc- 
trine. The question is, What is to be done in 
that event.? Answering out of the fullness 
of her own experience in such cases, Mathesis 
will venture to offer her sister the following 
counsel. "My years and station," she will 
say to Theology, "and the character of my 
occupation entitle me to believe that I am 
not without some insight into the nature of 
your gravest difficulty and not without some 
knowledge of the means available for over- 
coming it. JJsus, magister egregius, hoc me 
docuit. I, too, in the course of my long 



THE OLD THEOLOGY 115 

career have expended, I do not say have 
wasted, much time and energy in attempting 
to combine the non-combinable, in attempt- 
ing, that is, to erect a solid and unitary 
doctrine respecting some object of my 
thought upon a basis of postulates that 
were indeed individually sound and eligible, 
but that, taken collectively as a system, 
were subsequently found to involve logical 
incompatibility and so not to allow any 
superstructure not doomed to quick decay 
by the presence within it of fatal contra- 
dictions. Fortunately, I have not besought 
or trusted any hyperlogical providence to 
preserve such architecture against external 
criticism or the destructive agency of its 
own defects, but have had the grace to tear 
it down myself and prepare to build anew. 
My practice has been to examine again and 
patiently to reexamine the basal postulates, 
to form from them by trial and experiment 
as many subgroups as possible, subject to 
the condition that each of these be entirely 
free of interior inconsistence, and then, upon 
the ^i^bgroups as distinct though related 
foundations, to construct as many distinct 
but kindred doctrines, each of strength to 



116 THE NEW INFINITE AND 

mock at time and endure for aye. And my 
practice, as you and all the world may know, 
has been justified of its fruits. Examples 
abound in every division of my common- 
wealth, and some have come to fame. To 
cite but three of these — ^behold the noble 
structures of Euclid, of Bolyai and Lobat- 
schevski, and of Riemann. There stand the 
great geometries, each upon its own founda- 
tion of compatible postulates, and there, 
flawless within, unassailable from without, 
they will stand for ever, eternal witnesses 
of the fact that, contrary to many a ven- 
erated but shallow creed, one object of 
thought may, by virtue of its kind and not 
of limitations of the human mind, transcend 
the bounds of any one constructible theory, 
and in its own ultimate nature allow and 
validate at once, without annulling their 
differences, a class of dissonant doctrines. 
Thus you perceive, for example, that my 
Geometry is one, though my geometries are 
many — just as Music is one, though its 
forms be as varied as the moods of the sea. 
And I, Mathesis, am one, as Poetry is one, 
though my theories, my doctrines, are legion; 
for these but differ among themselves, as the 



THE OLD THEOLOGY 117 

myriad forms of Art : each is assertable, each 
being valid, of one great Form common to 
them all. My meaning, I trust, is clear. 
Conquest of your gravest difficulty demands 
division. By the method of trial and experi- 
ment, the fundamental attributes that you 
hypothetise of Deity must be assorted into 
sets each composed of harmonious elements. 
Implicit in each such group is a coherent and 
sacred doctrine. As these doctrines unfold, 
your conception of yourself will change : you, 
Theology, will indeed be one; but many 
your theologies. And thenceforth the Object 
of all your thought will appear to you and 
will be shown by you to the world, not in the 
light of a solitary sun, but in that of a 
constellation." 



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